SINGULARITIES AND INTEGRABILITY OF BIRATIONAL DYNAMICAL SYSTEMS ON PROJECTIVE PLANE

Simion Stoilow Institute of Mathematics Romanian Academy
HABILITATION THESIS
SINGULARITIES AND INTEGRABILITY OF BIRATIONAL
DYNAMICAL SYSTEMS ON PROJECTIVE PLANE
ADRIAN STEFAN CARSTEA
Specialisation: Mathematical Physics
Bucharest, 2013
1
Contents
1 Abstract
4
2 Rezumat
6
3 Overview
3.1 Role of singularities . . . . . . . . . . . . . . . . .
3.2 Integrable discrete systems . . . . . . . . . . . . .
3.2.1 Singularity Confinement . . . . . . . . . .
3.2.2 Complexity growth and algebraic entropy .
3.3 Deautonomisation . . . . . . . . . . . . . . . . . .
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4 QRT mapping
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4.1 The A1 matrices for various QRT mappings . . . . . . . . . . 28
5 Rational surfaces and elliptic fibrations
5.1 Discrete mappings and surfaces . . . . . . . . . . . . . . . . .
5.2 Preliminaries on rational elliptic surfaces . . . . . . . . . . . .
5.3 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . .
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6 Examples
6.1 Case ii-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Case i-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Case ii-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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7 Q4 mapping
52
7.1 Resolution of singularities and symmetry group . . . . . . . . 54
8 Minimization of elliptic surfaces from birational dynamics
8.1 Blowing down structure . . . . . . . . . . . . . . . . . . . .
8.2 A simple example which needs blowing down . . . . . . . . .
8.3 Discrete Nahm equations with tetrahedral symmetry . . . .
8.4 Discrete Nahm equations with octahedral symmetry: . . . .
8.5 Discrete Nahm equations with icosahedral symmetry . . . .
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9 Linearizable mappings
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9.1 A non-autonomous linearizable mapping . . . . . . . . . . . . 73
9.2 Discrete Suslov system . . . . . . . . . . . . . . . . . . . . . . 74
2
9.3
9.4
Other new linearisable systems . . . . . . . . . . . . . . . . . 76
Linearisable mappings of Q4 family . . . . . . . . . . . . . . . 80
10 Ultradiscrete (tropical) mappings
10.1 Ultradiscrete singularities and their confinement . . . . . . .
10.2 Nonintegrable systems with confined singularities and integrable systems with unconfined singularities . . . . . . . . .
10.3 A family of integrable mappings and their ultradiscrete counterparts . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.4 Complexity growth of ultradiscrete systems . . . . . . . . . .
10.5 Linearisable ultradiscrete dynamics: example from a biological
model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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11 General conclusions
103
12 Future research directions
104
3
1
Abstract
The main topic of this thesis is the singularity analysis and integrability of
two dimensional discrete dynamical systems (mappings). It is based essentially on the papers [1, 2, 3, 4, 5, 6, 7, 8] which posed the problem in the
context of autonomous dynamical systems. The main tools are based on
singularity confinement, algebraic entropy and their rigurous formulation using algebraic geometry of rational elliptic surfaces. It is an outcome of the
progress accomplished in the domain of discrete integrability which started
in 1990 with the introduction of singularity confinement and culminatig in
2001 with the definitive classification of discrete Painlev´e equations [57] using
generalised Halphen surfaces and affine Weyl groups.
Gradually it was realised that the methods o algebraic geometry (appearred for the first time in the pioneering work of Okamoto [54]) can be extended to analyse the integrability and symmetries/invariants of two dimensional mappings. The paradigmatic example is the so called QRT (QuispelRoberts-Thomson) mapping which is a very general birational dynamical
system possesing an invariant expressed by a ratio of biquadratic polynomials and which can be parametrised by Jacobi elliptic functions. Later on
it served as the fundamental skeleton on which the whole discrete Painleve
hierarchy has been erected.
In this thesis we shall be concerned with mappings different from QRT.
There are physical and mathematical motivation for this aspect. The physical motivation is that birational nonlinearity appears very often in biochemical and molecular biological models so the integrability and construction of
invariants are extremely important because they give a kind of global understanding of the phenomenon (unlike numerical simulations which give only
solution to an initial value problem). Mathematically the algebraic geometry
of discrete dynamical systems is done mainly for chaotic case (focusing on
construction of invariant measures, entropy etc. [11], [10]). We consider that
integrable discrete dynamical systems deserves the same treatment and we
believe that algebraic geometric methods will be extremely fruitfull.
In the chapter 3, we shall present an overview of the main instances
of integrability. The presentation is a physicist oriented one focusing on
examples and intuitive explanations. Various concepts and tools are briefly
discussed for both ODE’s and PDE’s. Then the case of discrete systems
is presented in particular underlining the role played by the confining of
singularities and algebraic entropy.
4
In the chapter 4 we present the original result [1] of integrating the general
asymmetric QRT mapping and chapter 5 (which is the integral reproducing
of [2])deals with the rigurous formulation of singularity confinement based on
rational elliptic surfaces. We present a classification o mappings that preserve
an elliptic fibration and exchange or not singular fibers during evolution. This
classification is illustrated with various examples in chapter 6.
In the chapter 7 we present the original result about Q4 lattice equation
[3]. Since it is a master equation for all soliton partial discrete equation
consistent around the cube we were interested to see how its travelling wave
reduction fits in our classification.
Chapter 8 is devoted to the original results about systems which can be
lifted to automorphisms of non-minimal rational elliptic surfaces [4]. By using blowing down structure we minimize them and show how this mappings
possesses invariants of higher order. Integrable discretisations of Nahm equations are among the analysed cases.
Chapter 9 presents our original results on linearisable systems [7], [8].
Here the power of algebraic geometry is limited since they have an infinite
number o singularities and accordingly an infinite number o blowing ups is
needed. However for some examples the blow down structure can be used and
linearisation is found. Other examples (including degenerations of Q4) are
analysed experimentally showing some peculiar aspects at deautonomisation
(the general presence of a free function).
Chapter 10 deals with ultradiscrete (tropical) mappings. It is shown that
practically here there is no efective integrability detector. The tropicalisation
o singularity confinement is critically analysed on various examples [5]. A nice
example [6] taken from a biological model is ending the chapter. Conclusions
(ch. 11) and possible new research directions (ch.12) are the last chapters of
this thesis
5
2
Rezumat
Subiectul principal al acestei teze este analiza de singularitati si integrabilitatea sistemelor dinamice discrete bidimensionale ( mappings ). Ea se
bazeaza pe lucrarile [1, 2, 3, 4, 5, 6, 7, 8] in care s-au analizat in special sisteme dinamice autonome (se discuta si deautonomizarea celor liniarizabile).
Principalele instrumente sunt metoda confinarii singularitatilor, metoda entropiei algebrice si formularea lor riguroasa, folosind geometria algebrica a
suprafetelor rational eliptice. Domeniul teoriei integrabilitatii discrete a inceput practic in 1990 (desi studii de ecuatii integrabile discrete au aparut
odata cu nasterea teoriei solitonilor in anii 70) prin introducerea metodei
confinarii singularitatilor si a culminat in 2001 cu clasificarea definitiva a
ecuatiilor Painlev´e discrete [57] folosind suprafetele Halphen generalizate si
grupurile Weyl afine .
Treptat, s-a observat ca metodele de geometria algebrica (aparute pentru
prima data n lucrarea de pionierat a Okamoto [54]) pot fi extinse pentru
a analiza integrabilitatea si simetriile/ invariantii sistemelor dinamice discrete bidimensionale. Exemplul paradigmatic este asa-numita ecuatie QRT
(Quispel - Roberts - Thomson), care este un sistem dinamic birational foarte
general, ce poseda un invariant exprimat printr-un raport de polinoame biquadratice si care poate fi parametrizat de functii eliptice Jacobi . Mai tarziu
acest exemplu a reprezentat scheletul fundamental pe care toata ierarhia
Painlev´e discreta a fost construita.
n aceasta teza vom discuta ecuatii diferite de QRT . Exista atit motivatie
fizica cit si matematica pentru aceasta. Motivatia fizica vine din faptul ca
neliniaritatea birationala apare foarte des n modele biologice, biochimice si
moleculare, astfel incit integrabilitatea si constructia invariantilor sunt extrem de importante, deoarece dau un fel de ntelegere globala a fenomenului
(spre deosebire de simularile numerice care dau doar o solutie la o problema
particulara de conditie initiala/de frontiera) . Cea matematica este legata
de faptul ca geometria algebrica se aplica cu succes in domeniul sistemelor
dinamice discrete haotice (cu precadere pentru calculul entropiei topologice
[11], [10] etc.). Noi consideram ca si sistemele dinamice discrete integrabile
merita acelasi tratament iar apliactiile geometriei algebrice s-au dovedit pina
acum fructuoase.
n capitolul 3, vom prezenta o imagine de ansamblu a principalelor concepte de integrabilitate. Prezentarea este una folosita in general de fizicieni
concentrandu-se pe exemple si explicatii intuitive. Diferite concepte si instru6
mente sunt discutate pe scurt atat pentru ecuatii diferentiale cit si pentru
cele cu derivate partiale . Apoi, cazul sistemelor discrete este prezentat mai
in detaliu subliniind rolul jucat de singularitati si entropiea algebrica .
n capitolul 4 vom prezenta rezultatul original [1] legat de integrarea directa a sistemului QRT asimetric; capitolul 5 ( care este reproducerea integrala a lucrarii [2] ) se ocupa cu formularea riguroasa a confinarii de singularitati bazata pe teoria suprafetelor eliptice rationale . Vom prezenta o
clasificare a transformarilor care invariaza o fibrare eliptica si schimba sau
nu fibrele singulare in timpul evolutiei. Aceasta clasificare este ilustrata cu
exemple variate in capitolul 6.
n capitolul 7 vom prezenta rezultatele originale despre ecuatia discreta
Q4 [3] . Avnd n vedere ca este o ecuatie master pentru toate ecuatiile solitonice partial discrete cu consistenta cubica am fost interesati sa vedem cum
reducerea de tip “unda progresiva” se incadreaza in clasificarea noastra .
Capitolul 8 este dedicat rezultatelor originale cu privire la sistemele care
pot fi ridicate la automorfisme ale suprafetelor eliptice relativ non-minimale
[4]. Prin utilizarea structurii de “blow-down” (eclatare inversa sau contractie)
suprafata se minimalizeaza lucru care conduce in mod remarcabil la invarianti
de ordin superior . Discretizarile integrabile dale ecuatiilor reduse Nahm din
teoria cimpurilor de etalonare sunt printre cazurile analizate .
Capitolul 9 prezinta rezultatele noastre originale cu privire la sistemele
linearisable [7],[8]. Aici metodele de geometrie algebrica nu mai merg, deoarece
aceste sisteme au un numar infinit o singularitati si in consecinta este necesar un numar infinit de eclatari (blow ups). Cu toate acestea, pentru unele
exemple daca se contarcta anumite clase de divizori exceptionali gasiti direct
procedeul de liniarizare rezulta imediat. Alte exemple (inclusiv degenerari
ale Q4) sunt analizate experimental precum si o serie de aspecte particulare
care apar la deautonomisare (prezenta generala a unei functii arbitrare in
definirea coeficientilor) .
Capitolul 10 se refera la sistemele ultradiscrete (tropicale) . Se arata
ca practic, aici nu exista nici un detector de integrabilitate. Tropicalizarea
metode confinarii singularitatatilor este analizata critic pe diverse exemple
[5]. Capitolul se incheie cu un exemplu interesant de sistem partial ultradiscret liniarizabil venit dintr-un model biologic [6]. Concluzii (cap.11) si
posibile noi directii de cercetare (cap. 12) sunt ultimele capitole ale acestei
teze
7
3
Overview
The nonlinear science is coping today with a very deep problem; can one
single out and describe to a certain degree of accuracy the complexity and
self-organization exhibited by a nonlinear dynamical system? This fact imposes the next question related to existence or non existence of some amount
of hidden symmetry which would help in decribing that. Today these problems are still open, despite many deep results obtained so far. In this context,
the integrable nonlinear dynamical systems play a special role. First of all,
the fact that they are integrable might give the impression that they are
not very important since they are very rare. Indeed! The great majority
of dynamical systems emerging from models (in physics, biology, economy
etc) are not integrable and chaotic. But on the other hand, integrability exhibits a huge amount of hidden symmetry in various ways. This in turn gives
rich structure which can be described in a clear and accurate way. It turns
out that many problems with unexpected structure and self-organization are
related in some way to integrable systems. Roughly speaking, nonlinear dynamical systems are “rules of evolution” for given quantities subjected to
self-interaction (otherwise the dynamics would have been linear and not interesting). These rules can be put in a form of a nonlinear differential/discrete
equations (ODE/O∆E), partial differential/discrete (PDE/P∆E) equations
or cellular automata (CA) (which are also discrete equations with the dependent variables having values in a denumerable or finite field).
Now, what is integrability? Given a nonlinear finite or infinite dimensional dinamical system (continuous or discrete) when can we say that it is
completely integrable? This is a question with no clear cut answer. And this
is because there are many characterisation of integrability. First one which
seems to be quite intuitive is related to the possibility of finding a solution
with enough number of free constants such that this solution could be considered general. However this approach can somehow be misleading. Lets
take the example of the famous logistic map which exhibits chaotic behaviour
through period doubling in cascade birfucations [12]
xn+1 = 4xn (1 − xn )
(3.1)
The coefficient 4 in front of the RHS of the equation places the dynamics
in the fully chaotic region. However the equation can be solved analytically,
8
namely the solution is given by:
1
xn = (1 − cos(2n c0 ))
2
which depends on an arbitrary constant c0 (fixed by the initial condition). So
one can say that the system has a general solution but still the system is in the
chaotic region. This can be easily seen from the “buterfly effect” (exponential
growth related to the small variation of initial condition) namely:
dxn
= 2n−1 cos(2n c0 )
dc0
Accordingly, the definition of integrability must be somehow posed in
a different setting. A more appropriate way to characterize the integrability would be not relying on the exact solutions but rather on the global
information given by the integrals (or invariants or conservation laws) and
symmetries (we shall define later what means the symmetry). The first definition of integral has been given by Darboux and Goursat [13]. They say that
“an integral is general and useful if provides all the arbitrary data needed for
expressing the solution (whose existence must be proved or it is guaranteed by
the Cauchy theorem)”. In the same spirit, according to Poincar´e, to integrate
a differential equation is to find, for the general solution, a finite expression,
possible multivalued, in terms of a finite number of functions. This definition
has a very important connection with the concept of singularity as we shall
see in the next part.
In order to clarify the integrability concept we are going to give a very
brief description of main types of integrability used for ODE’s/PDE’s and
then to discrete systems. This review is based on the informations found in
[65, 66]
The first type of integrability is the so called “integrability by quadratures”. For instance, if we take the nonlinear ordinary differential equation:
x¨ = ax + bx2 + cx3
the integral is given by:
x˙ 2 ax2 bx3 cx4
−
−
−
2
2
3
4
and the integration is realised by the quadrature
Z
dx
p
t − t0 =
2
2(I + ax /2 + bx3 /3 + cx4 /4)
I=
9
The integral can be easily computed using Jacobi elliptic functions. Since I
is known by the initial condition the solution is readily obtained by inversion.
This type of integrability is rather restrictive. For instance the equation
x˙ + x2 + t = 0
cannot be integrated by quadratures since it is non-autonomous. However
it is integrable by direct linearisation since we use the substitution x(t) =
˙
y(t)/y(t)
then we get
y¨(t) + ty(t) = 0
which is solvable by Airy functions (this is somehow tautological since the
Airy functions are defined by such a liniar ODE
The concept of integral is well known and it is applied widely specially
in the hamiltonian mechanics (the above example is a particular one dimensional hamiltonian system). For an N dimensional hamiltonian system
the concept of integrability is clear. Because of the symplectic structure of
phase space the identification of N invariants of motion (integrals) allows
simplification to a set of trivial integrations (equations of motion in the so
called action-angle variables). This type of integrability is called Liouville
integrability.
The existence of integrals is not limited only to hamiltonian (conservative)
systems. Even strongly dissipative systems can be integrated using the so
called time-dependent integrals. For instance, the famous Lorentz system
[14]
x′ = σ(y − x)
y ′ = ρx − y − xz
z ′ = xy − bz
The following values of the parameters make the system integrable [15]:
i)σ = 0
ii)σ = 1/2, b = 1, ρ = 0
iii)σ = 1, b = 2, ρ = 1/9
iv)σ = 1/3, b = 0, ρ = free
In the case:
i) the system is linear
10
ii) it has two time dependent integrals namely [16]:
I1 = (y 2 + z 2 )e2t , I2 = et (x2 − z)
which reduces the system to a quadrature and the solutions can be expressed
in terms of elliptic functions.
iii) we do have one time dependent integral I3 = e2t (x2 − 2z) which turn the
system, after a change of variables, into the Painleve II equation.
iv) can be combined in a third order equation which can be integrated to
give the following time dependent integral:
I4 = e4t/3 (x¨
x − x˙ 2 + x4 /4)
Changing variable to X = xet/3 and T = e−t/3 transform the above equation
in the Painleve III equation
1
X ′2 X ′
−
+ X3 −
X
T
X
where the new variables are defined through:
X ′′ =
x(t) =
(3.2)
2ic −t/3
e
X(T ), T = ce−t/3 , c4 − (3K)4 = 0
3
The most powerfull type of integrability is the so called integrability by
Lax pair or by spectral methods. The idea is to write the nonlinear ODE
under consideration as a compatibility condition of two linear operators and
to move the whole analysis in a different space where everything is linear
(in the case of ODE in the space of monodromy data). Then, by inverse
spectral transform one can obtain the solution using some singular integral
linear equations (which cannot be solved in a closed form but still provides
a lot of informations about solutions and their asymptotology)[17]
In the domain of PDE’s the situation is somehow similar. We do have
here direct linearisation, for instance in the case of two dimensional NavierStokes equation (Burgers equation) ut + uux + uxx = 0. The Cole-Hopf
transform u = ∂x log F will turn it into the heat equation Ft + Fxx = 0.
Finding integrals for a given PDE is a rather complicated procedure since
any PDE is an infinite dimensional dynamical system. Accordingly, complete
integrability would require infinite number of integrals. It turns out that this
is the case for the so called soliton equations which can be cast in the so
11
called Lax representation. For example, we take the famous Korteweg de
Vries equation which is a paradigmatic soliton equation:
ut + 6uux + uxxx = 0
It can be written in the following form [18]:
ut + 6uux + uxxx =
∂L
− [L, B] = 0
∂t
where the operators L = ∂x2 + u(x, t) and B = −4∂x3 − 3{∂x , u} are called
Lax pairs. There is no algorithm for finding such operators and moreover
they are not unique, making problem very hard. Since B is an antisymmetric
operator, the spectrum of L is invariant in time. So the problem of initial
condition will fix the spectral data and then, using the inverse method in
the scattering theory associated to L, the solution can be computed at any
moment of time by means of a linear integral equation. In addition, the
integrals of motion can be computed by traces of powers of operator L and
every integral of motion can be considered as a hamiltonian generating a
flow. The family of such flows (equations) forms the so called KdV-hierarchy.
In this way the Liouville integrability is intimately related to the existence
of Lax operators [19]. Related to this method is another vey interesting
aspect specific to integrable PDE’s namely the bihamiltonian structure. For
example, KdV equation it can be written easily in the hamiltonian form:
Z
δ
u2x
3
ut = J 0
≡ J0 δ u H 1
dx u −
δu R
2
d
where the symplectic operator is J0 = dx
. In the late seventies Magri discovered a remarkable fact that it can be written in a different way using a
different symplectic form
2
Z
d
d
δ
u
d
+ 2u + 2 u
dx
≡ J1 δ u H 0
ut =
3
dx
dx
dx
δu R
2
The main consequence of this aspect is the possibility of generating the whole
KdV hierarchy using the recursion operator R = J0−1 J1 through:
utn = J0 δu Hn+1 = J1 δu Hn
There is also another type of integrability which is specific to PDE namely
Hirota integrability or integrability by Hirota bilinear method. This method
12
has been introduced by Hirota in 1971 [20] and it says that if a quasilinear
hyperbolic equation has a general N -soliton solution for any N then the
equation is completely integrable. Physically speaking N -soliton solution
means that multiple collisions of arbitrary solitons are allowed. The main
advantage of this method is that it is a direct one and can be applied to any
equation (continuous, discrete or semidiscrete). Again we take the KdV case
for illustration:
ut + 6uux + uxxx = 0
(3.3)
and we put u(x, t) = 2∂x2 log F (x, t). Then our KdV will transform into a
more complicated equation but bilinear and quadratic:
2
F Fxt − Fx Ft + F Fxxxx − 4Fx Fxxx + 3Fxx
=0
which can be written as:
(Dt Dx + Dx4 )F · F = 0
(3.4)
The bilinear antisymmetric operator D is given by:
Dxn a(x) · b(x) := (∂x − ∂y )n a(x)b(y)|x=y
(3.5)
So in this way the nonlinearity of the original KdV equation has been swallowed and the bilinear equation shows practically the dispersion relation
of the linear part of the KdV (this can be easily seen if we formally put
Dt → ω, Dx → k). We have to emphasize that this operator appeared also
long time ago in the papers of Borel and Chazy [21], [22] where they showed
that equations written with this operator have solution which are complex
entire functions.
Hirota proved that the N -soliton solution of the KdV equation can be
written in terms of exponentials for the function F and has the following
expression:
!
N
X
X
X
F (x, t) =
µi (ki x − ωi t) +
µi µj Aij
(3.6)
exp
µ1 ,...,µN ∈{0,1}
i=1
i<j
where ωi = −ki3 is the dispersion relation of the linearised equation and
Aij = ((ki − kj )/(ki + kj ))2 is the interaction term between the soliton i and
soliton j.
13
The method can be applied to any equation and it was observed that very
few equations possesses N -soliton solution. The great majority have one and
maximum 2-soliton solution (namely N = 2 in the above formula (3.6)). On
the other hand it was also observed that once 3-soliton solution is allowed
then automatically N -soliton solution is as well (this is still a conjecture) and
these equations are precisely those which are completely integrable. This fact
has been studied thoroughly in the middle of eighties in the papers of Jimbo
and Miwa who showed the deep algebraic meaning of Hirota integrability
[23](bilinear hierarchies are related to vertex operators representations of
affine Lie algebras).
The importance of Hirota integrability relies on the fact that it applies
equatlly to discrete and differential-discrete equations and, moreover makes
connection with the role of singularities in the working definition of integrability. In the next chapter we shall discuss the role of singularities:
3.1
Role of singularities
We have seen that in the definition of Poincar´e solution means a finite expression possible multivalued in terms of a finite number of functions. When
we discussed integrability by linearisation we encounter the special functions
(Airy function in the example). However any special function is defined by a
linear differential equation which is better studied when we extend the analysis to complex domain. The importance of the analysis in the complex space
has been given in [24] where the analysis of a purely real power spectrum of
a signal has been done. It was shown that the high frequency behaviour of
the Fourier transform depends on the location and nature of singularities in
the complex time plane.
Also, thanks to modern analytic theory in the complex domain, the global
information for an ODE can be obtained by an analytic continuation of locally defined solutions. Now, we have to fix the statements here; A singular
point is a point which breaks the analyticity of a solution for an ODE. If
there is multivaluedness in its neighbourhood, then the singularity is critical,
or sometimes is called branched singularity. If one wish to define a function
there is a requirement to treat singular points such that to restore the singlevaluedness. This can be done by the so called uniformisation (by introducing
contours, Riemann surfaces etc.). This procedure can be always done in the
case of solutions of linear ODE, and this is possible because the location of
singularities is fixed - namely are completely determined by the coefficients.
14
So, according to Poincar´e definition every solution of a linear ODE defines
a function and any linear ODE is an integrable system.
The nonlinear ODE lose this property because the location of singularities
depends on the initial conditions (or equivalently the integration constants).
So defining a function as a solution of a nonlinear ODE becomes a hard problem. Based on the results of Kowalevskaya [25], Fuchs and Painlev´e defined
the so called Painlev´e property and moreover they were able to construct
the most general order two nonlinear ODE’s which define new special functions beyond the elliptic ones, namely Painlev´e transcendents [26]. In a few
words, the Painlev´e property imposes that the movable singularities (meaning that they depend on the initial conditions) of a given nonlinear ODE in
the complex plane, be at most poles. This fact places the dynamics of the
considered ODE, on the Riemann sphere which is a compact and “regular”
object and, accordingly, it is considered to be compatible with a smooth, predictable dynamics (integrability). On the other hand, presence of branching
and essential singularities would proliferate the number of Riemann sheets
and the evolution is no longer “integrable”. We must stress on important
point here. It is considered that Painlev´e property is not just a predictor of
integrability but practically a definition of integrability. As such it becomes
rather a tautology than a criterion. This is the case because in the eighties
it was discovered that practically all integrable soliton PDE’s when reduced
they become equations which obey Painleve property. But on the other hand
it is crucial to make distinction between Painleve property and various algorithms for investigation (like Painlev´e test for instance which search for
movable branch points subject to certain assumptions). There is no algorithm so far that guarantee Painlev´e property. However the application of
these algorithms (mainly Painlev´e test) gave many interesting results even
to chaotic ODE’s and PDE’s [27] . Still there are systems which are solvable
(by quadratures and cascade linearization) and are not related to singularity
structure. Accordingly the Painlev´e property is not always equivalent with
solvability.
The key steps in application of the Painleve test are the following: Suppose we start with a system of nonlinear ODEs:
w˙ i = Φ(w1 , w2 , ..., wn ; t)
(3.7)
Then the main idea is to see the asimptotology of a solution around a singularity. For instance if t0 is a generic point one tries to see the dominant
15
behaviour of the solution in the form:
wi = ai (t − t0 )pi
where some reals,parts of pi are negative. Substituting in the sistem and
relying on the maximals balance principle [49] one can find possible dominant
behaviours. If one of the pi is noninteger then we are in the situation that
t0 is a movable branch point which is incompatible with Painlev´e property
so in this case our system of ODE is likely to be not integrable (further
refinements can be done in the terms of the so called weak Painlev´e property
but we shall not dwell on this). If all pi are integers then for each of them
then the leading behaviour can be seen as the dominant term of a Laurent
series around a movable pole:
wi = (t − t0 )
pi
∞
X
0
(m)
ai (t − t0 )m
(3.8)
(0)
where ai = ai and the location of t0 is the first integration constant. The rest
(m)
of n − 1 constants are among the coefficients ai and if their corerspondent
powers m are integers as well then the system is free of any branching and
from the existence of n constants of integration, it is likely to be a completely
integrable one. However this is just a necesary condition and moreover it does
not capture the presence of essential movable singularities. Further investigation are necessary to establish the sufficiency by constructing integrals or
Lax pairs.
The bad role of branching of singularities can be grasped by the following
very simple ODE [27]:
dx(t)
A
B
C
=
+
+
dt
t−a t−b t−c
(3.9)
Its integration by quadratures gives:
I = x(t) − A log(t − a) − B log(t − b) − C log(t − c)
(3.10)
It well known that in the complex plane the logarithm is defined up to an
integer multiple of 2iπ so the integral (3.10) is determined up to the term
2iπ(kA + mB + nC) with k, m, n ∈ Z. Now if one or two of the A, B, C
are zero one can construct a two or one dimensional lattice and define I in
16
a unique way. But if ABC is not zero and A, B, C are linearly independent
over the integers then we have a big indeterminacy in constructing I beacuse
its value can fill densely the whole plane. So the integral is not useful if we
have dense multivaluedness and accordingly the above ODE is not integrable
in this case.
Anyway one can argue that practically any dynamical system has an
integral, namely the initial condition. For instance if we consider the sistem
of ODE:
x˙ i = Fi (t, x1 , ..., xn ), i = 1, ..., n
(3.11)
with initial condition xi (t0 ) = ci . The general solution of the system is:
xi (t) = fi (t, c1 , ..., cn )
and by inverting it we get ci = Ii (t, x1 , ..., xn ) which we can consider to be
the integrals. However this inversion is not at all guaranteed to be single
valued.
As we have said Painleve analysis has been thoroughly applied to nonlinear ODE and PDE’s and we are not going to insists here. We shall concentrate mainly on discrete systems.
3.2
Integrable discrete systems
In the case of discrete systems the problem is completely different. Since
now we have practically a recurence relation and everything is not local it is
impossible to apply the instruments of complex analysis (expansions around
singularities since now there are not neighbourhoods at all). In addition, it is
impossible to “discretise” the results from continuous systems because there
are many (in fact infinity) ways to discretise a continuum system and also a
discrete system can have many continuum limits. Many of the properties of
a continuous system are not at all preserved by the discretisation procedure.
For instance, the Riccati equation
x′ = ax2 + bx + c
(3.12)
can be discretised either as writing the derivative as finite difference
xn+1 = ax2n + (b + 1)xn + c
17
(3.13)
or adding also some factor to the nonlinear term
xn+1 − xn = axn xn+1 + bxn + c ⇐⇒ xn+1 =
(b + 1)xn + c
1 − axn
(3.14)
There is a huge difference between (3.13) and (3.14). The first one is a logistic type mapping which is fully chaotic and the second is a homographic
mapping which is integrable by a Cole-Hopf transform. So only the second
discretisation preserve the properties of the initial continuum Riccati equation (3.12).
Because of these many ways to discretize a dynamical system one needs a
tool to detect at least the necessary conditions for integrability. By integrability in the discrete case we understand the same as in the continuos one,
namely for a k-dimensional discrete nonlinear system we need [65]:
• existence of a sufficient number of integrals or conservation laws expressed as rational expression Fk (xn , xn+1 , ...xn+k ) invariant under the
action of the mapping (however defining a hamiltonian structure is not
guaranteed)
• possible linearisability by some transformations of dependent variables
(like the above Cole-Hopf)
• existence of a Lax pair
• existence of general multisoliton solution in the infinite dimensional
case
3.2.1
Singularity Confinement
A very efficient tool in detecting possible candidates for integrability is the
so called singularity confinement test discovered in 1991 by A. Ramani, B.
Grammaticos, V. Papageorgiou [28]. The idea has roots in the Painlev´e
analysis for continuos systems. As we have seen in the integrable case the
singularities are just poles. In the nonintegrable case sigularities may accumulate in fractal boundaries (so a natural boundary appears). Here in the
discrete setting the analysis is not based on Laurent expansion but rather
on the behaviour of iterations in some movable singular points. More precisely, if the mapping leads to a singularity (depending on initial conditions)
then after a finite number of steps (iterations) the singularity must dissapear
18
(confinement) without loss of information of initial condition. Thus the confinement is reminiscent to absence of natural boundaries (where singularities
accumulate) in integrable continuous systems. On the other hand, preserving
of information of initial condition is in contrast with chaotic dynamics where
strange/fractal attractors absorb initial information. In order to implement
practically the criterion let us see how it works on a given example:
xn+1 + xn + xn−1 =
a
+b
xn
(3.15)
and suppose that starting with a given initial condition namely x−1 = f
(where f is an arbitrary complex number) we hit at the next iteration x0 = 0.
Now let us see what happens further on:
• x−1 = f
• x0 = 0
• x1 = −0 − f + a/0 + b = ∞
• x2 = −∞ − 0 + a/∞ + b = −∞
• x3 = ∞ − ∞ − a/∞ + b =?
So one can see that the emergent infinities (which are just apparent singularities since they can be treated as nonsingular in the projective space) lead
in the expression of x3 to a real singularity given by the ambiguity of ∞ − ∞.
To cope with this situation we use the argument of continuity with respect
to initial conditions and consider x−1 = f and x0 = ǫ and then expand in
power series of ǫ. We get:
• x−1 = f
• x0 = ǫ
• x1 = aǫ−1 + b − f − ǫ
• x2 = −aǫ−1 + f + ǫ +
f −b 2
ǫ
a
+ O(ǫ3 )
a
• x3 = aǫ−1 − f − ǫ + f −b
ǫ2 − aǫ−1 − b + f + ǫ + −aǫ−1 +f
+ b = ǫ + O(ǫ2 )
a
+O(ǫ)
• x4 = f + O(ǫ)
19
so the ambiguity is resolved and the initial information is recovered. Accordingly, the mapping is a possible candidate for an integrable one and indeed
the mapping can be integrated in terms of elliptic functions being in fact an
autonomous limit of a discrete Painlev´e equation. This integrability detector
can be applied also to partial discrete equations and what is really interesting it has a closed connection with Hirota bilinear formalism and existence
of multisoliton solution [29].
In our case we have seen that the singularity pattern (the sequence of
values of xn from initial condition up to its recovery) is (f, 0, ∞, ∞, 0, f ) so
it suggests that we can express xn using an entire function through:
xn =
Fn−1 Fn+2
Fn Fn+1
(3.16)
Now it is convenient to work with the discrete derivative of (3.15) namely:
xn+2 +xn +xn+1 −
a
xn+1
−b−xn+1 −xn −xn−1 +
a
a
a
+b = xn+2 −xn−1 −
+
=0
xn
xn+1 xn
Introducing (3.16) we get
2
Fn−1 (Fn+4 Fn − aFn+2
) = Fn+3 (Fn+2 Fn−2 − aFn2 )
which gives immediately the following Hirota bilinear form
Fn+2 Fn−2 − aFn2 − Fn+1 Fn−1 := (exp 2Dn − exp Dn − a)F · F = 0
where we have used the Hirota bilinear operators introduced in (3.5). The
solution of the bilinear equation is an entire function and it can be shown
that is given by Riemann theta function (in fact the bilinear equation is
nothing but a particular case of the famous Fay identitiy for Riemann theta
functions)
3.2.2
Complexity growth and algebraic entropy
Unfortunately the singularity confinement test is just a necesary condition.
This can be seen by the famous counterexample the so called HietarintaViallet mapping [30]:
1
(3.17)
xn+1 + xn−1 = xn + 2
xn
20
It has the following singularity pattern (f, 0, ∞2 , ∞2 , 0, f ) so we do have confining. However a pathology can be seen in the numerical simulation of the
equation which shows fully developed chaos. So a new stronger criterion is
needed. Here we give the most powerfull integrability criterion namely the
algebraic entropy. It is based on the notion of complextity introduced by
Arnold [31] which is the number of intersection points of a fixed curve with
the image of a second curve obtained under the iteration of the mapping.
This idea has been extended by Viallet and colaborators who introduced the
idea of algebraic entropy which encodes globally the complexity by means
of degrees of iterates. More precisely, if a birational mapping starts with
a polynomial degree (of numerator or denominator) d then the n − th iterate will have the degree dn . When the mapping is integrable, some strong
simplifications occur and the degree growth is polynomial in n instead of
exponential. Let us illustrate on the example given by the mapping (3.15).
Suppose x0 = p and x1 = q/r (which practically means iteration of numbers
in the projective space). Then the following sequence of polynomial degree
of the denominator we have [32]:
1, 2, 4, 8, 13, 20, 28, 38, 49, 62, 76...
which can be fitted by the formula,
1
dn = (9 + 6n2 − (−1)n )
8
where n is the iteration. So clearly the growth is polynomial in accord with
the integrability. On the other hand the Hietarinta-Viallet mapping which is
confining but chaotic has the following sequence of degrees:
0, 1, 3, 8, 23, 61, 162, 425...
and the degree growth obeys the recursion relation dn+4 = 3(dn+3 −dn−1 )+dn .
This gives the expression of algebraic entropy
S = lim log dn /n
n→∞
√
which in this case is S = (3 + 5)/2. A nonzero algebraic entropy is the
sign of chaos. The zero algebraic entropy (or equivalently the polynomial
degree growth) is the detector of integrability. The singularity confinement
and algebraic entropy were proved to be instrumental in the majority of
analysis of discrete systems. The discovery and properties of discrete Painlev´e
equations relies on them.
21
3.3
Deautonomisation
Another important aspect of singularity confinement and complexity growth
is the procedure of deautonomisation [29]. It means that for a mapping
one can put coefficients to depend on the independent variable and still the
mapping to be integrable. The method is quite simple namely to impose the
same singularity pattern (or complexity growth) for both autonomous and
nonautonomous mapping. This will result in a constraint on coefficients. Let
us illustrate on the same mapping (3.15) but having a = a(n)
xn+1 + xn + xn−1 =
an
+b
xn
(3.18)
So the singularity confinement is:
• xn−1 = f
• xn = ǫ
• xn+1 = an ǫ−1 + b − f − ǫ
• xn+2 = −an ǫ−1 + f + ǫ +
an+1 /an (f −b) 2
ǫ
an
n+1 +an
• xn+3 = − an+2 +a
ǫ + ( an+1
b−
an+2
an
+ O(ǫ3 )
an+1 +an+2
f )/an ǫ2
an
+ O(ǫ3 )
n+2 −an+1 +an an
+ O(ǫ)
• xn+4 = − an+3an−a
+an+1 +an+2
ǫ
And indeed for generic an xn+4 is still divergent. But if we impose that
xn+4 to have the same expression as in the autonomous limit then we get the
following discrete linear equation for an namely:
an+3 − an+2 − an+1 + an = 0 =⇒ an = αn + β + γ(−1)n
and our mapping becomes:
xn+1 + xn + xn−1
αn + β + γ(−1)n
=
+b
xn
with α, β, γ are free constants. The resulting mapping is nothing is nothing
but the discrete Painlev´e I or II equation. The name comes from the continuous limit; namely if γ = 0 then the continuous limit can be computed as
follows: Consider t = ǫn and xn = w0 (t) + ǫw1 (t) + ǫ2 w2 (t) + O(ǫ3 ) where
wi (t) are unknown functions. For the shifted variable xn+1 the functions
22
wi (t) appear with shifted argument as well but we expand them in Taylor
series namely:
wi (t + ǫ) = wi (t) + ǫwi′ (t) +
ǫ2 ′′
w (t) + ...
2 i
Also we need b = b0 + ǫb1 + ... and an = a0 (t) + ǫa1 (t) + ... The difficult part
is that we do not know which of the ai are constant or not and which of the
wi (t) is the dependent variable. This requires a lot of “intuition” so that is
why an equation can have many continuous limits. In our case if we take
xn = 1 +2 w(t), an = −3 − ǫ4 t, b = 6 then in the limit ǫ → 0 we get:
w′′ (t) + 3w2 (t) + t = 0
which is the Painleve I equation.
For γ not zero the equation has bigger freedom and can be written as an
asymmetric system. Namely if Xm = x2m , Ym = x2m+1 then we have:
Ym + Xm + Ym−1 =
Xm+1 + Ym + Xm =
2αm + β + γ
+b
Xm
2αm + α + β − γ
+b
Ym
Now if
Xm = 1 + ǫw + ǫ2 u, Ym = 1 − ǫw + ǫ2 u, 2αm + β = 1 − ǫ3 m, γ = −ǫ3 c/4
then we get
1
u = (w2 − w′ + t)
4
leading to
w′′ − 2w3 − 2tw − c = 0
which is the Painlev´e II equation. The symmetric and asymmetric notion of
the mapping will be clarified in the next section when we discuss QRT mappings. The deautonomisation procedure was deeply investigated in connnection with the theory of discrete Painlev´e equations and their properties. We
are not going to discuss this topic since it is too vast. Rather we shall
focus on the rigurous aspects of singularity confinement using tools from
algebraic geometry which in turn will help not only to establish the integrable/nonintegrable character but also to integrate effectively any mapping
by computing invariants.
23
4
QRT mapping
The basic object in the study of integrability of two dimensional mappings
is the so called QRT system. It was introduced in the beginning of nineties
by Quispel, Roberts and Thomson [33] and by now is considered to be the
paradigm of discrete integrability (in 2010 a whole book appeared dedicated
to QRT mappings [33]). The importance o this system relies on the act that
it gives a rather general discrete equation with a solution written in terms o
elliptic function and possessing a biquadratic invariant.
There exist two families of QRT mappings [29], [33], [34], [35] which
are dubbed respectively symmetric and asymmetric for reasons which will
become obvious below. One starts by introducing two 3 × 3 matrices, A0 and
A1 of the form


αi βi γi
(4.1)
Ai =  δi ǫi ζi 
κ i λi µi
If both these matrices are symmetric the mapping is called symmetric.Oth
x2
~ = x
erwise it is called asymmetric. Next one introduces the vector X
1
 
 
g1
f1
~
~



and constructs the two vectors F ≡ f2 and G ≡ g2  through
g3
f3
~ × (A1 X)
~
F~ = (A0 X)
(4.2)
~ = (A˜0 X)
~ × (A˜1 X)
~
G
where the tilde denotes the transpose of the matrix. The components fi , gi
~ are, in general, quartic polynomial of x. Given the fi , gi
of the vectors F~ , G
the mapping assumes the form:
xn+1 =
f1 (yn ) − xn f2 (yn )
f2 (yn ) − xn f3 (yn )
(4.3)
g1 (xn+1 ) − yn g2 (xn+1 )
(4.4)
g2 (xn+1 ) − yn g3 (xn+1 )
In the symmetric case we have gi = fi and (4.3), (4.4) reduces to a single
equation
f1 (xm ) − xm−1 f2 (xm )
xm+1 =
(4.5)
f2 (xm ) − xm−1 f3 (xm )
yn+1 =
24
with the identification xn → x2n , yn → x2n+1 .
~ are obtained as vector products it is clear that the reSince F~ and G
sult will be the same if one replaces the matrices A0 and A1 by the linear
combinations ρ0 A0 + σ0 A1 and ρ1 A0 + σ1 A1 where ρ0 , σ0 , ρ1 , σ1 are four free
parameters (with the only constraint ρ0 σ1 6= ρ1 σ0 ). This transformation can
be used in order to reduce the effective number of the parameters of the system to 14 in the asymmetric case and to 8 in the symmetric one. However
this is still not the number of the effective parameters since we have the full
freedom of a homographic transformation, which amounts to three parameters, separately for x and y in the asymmetric case and just for x in the
symmetric one. Thus the final number of genuine parameters in this system
is 8 for the asymmetric mapping and 5 for the symmetric one.
The QRT mapping possesses an invariant which is biquadratic in x and
y:
(α0 + Kα1 )x2n yn2 + (β0 + Kβ1 )x2n yn + (γ0 + Kγ1 )x2n + (δ0 + Kδ1 )xn yn2
+(ǫ0 +Kǫ1 )xn yn +(ζ0 +Kζ1 )xn +(κ0 +Kκ1 )yn2 +(λ0 +Kλ1 )yn +(µ0 +Kµ1 ) = 0
(4.6)
where K plays the role of the integration constant. In the symmetric case
the invariant becomes just:
(α0 + Kα1 )x2n+1 x2n + (β0 + Kβ1 )xn+1 xn (xn+1 + xn ) + (γ0 + Kγ1 )(x2n+1 + x2n )
+(ǫ0 + Kǫ1 )xn+1 xn + (ζ0 + Kζ1 )(xn+1 + xn ) + (µ0 + Kµ1 ) = 0
(4.7)
Viewed as a relation between xn and yn equation (4.6) is a 2-2 correspondence (and similarly for (4.7). While the generic biquadratic correspondence
is not in general integrable [36], leading to an exponential growth of the
number of images and preimages of a given point, this is not the case for
(4.6). (The symmetric case (4.7) is a well-known exception to this, being
indeed integrable). As a matter of fact it was argued in [29], due to the
specific structure of the mapping the correspondence (4.6) leads to just a linear growth of the number of images of a given point. Further results which
strengthen the integrability argument of (4.6) are the analyses presented in
25
[37] and [38]. As we have shown in [37] both symmetric and asymmetric
QRT mappings pass the singularity confinement test. Moreover in the case
of the symmetric mapping we were able to show that, within a given generic
singularity pattern, the QRT mapping was the only one to satisfy the singularity confinement criterion. The degree growth of the iterates of some
initial condition was studied in [38], using algrebraic entropy techniques. We
have shown there that both symmetric and asymmetric mappings have a zero
algebraic entropy and in fact lead to quadratic degree growth. We turn now
to the explicit integration of the QRT mapping. In the symmetric case the
integration (which, according to Veselov [39], is due to Euler) is presented
in a pedagogical way by Baxter [40]. Still we present below the details of
the calculation since they will help understanding the asymmetric case. So,
we start with the symmetric case and work with the integrated form. To
begin with, we drop the explicit reference to the parameters of the A0 and
A1 matrices and to the integration constant K and rewrite (4.7) as
αx2 y 2 + βxy(x + y) + γ(x2 + y 2 ) + ǫxy + ζ(x + y) + µ = 0
(4.8)
(We shall, of course, return to the explicit consideration of the A0 , A1 parameters and K). We introduce a homographic transformation x = (aX +
b)/(cX +d) (and the same for y which, in the symmetric case is just x shifted
by one step). Moreover we take d = 1, since we are not looking for a linear
transformation (neither in x nor in 1/x), and put a = sc. We demand that
the coefficient of the XY (X + Y ) and of (X + Y ) terms vanish and also that
the coefficient α of the X 2 Y 2 term be equal to the constant term µ. From
the latter we obtain:
c4 =
αb4 + 2βb3 + (2γ + ǫ)b2 + 2ζb + µ
αs4 + 2βs3 + (2γ + ǫ)s2 + 2ζs + µ
(4.9)
From the vanishing of the coefficient of the (x + y) term we find:
s=−
βb3 + (2γ + ǫ)b2 + 3ζb + 2µ
2αb3 + 3βb2 + (2γ + ǫ)b + ζ
(4.10)
Requiring as a last constraint that the coefficient of the XY (X + Y ) term
to vanish we obtain an equation for b which factorizes into a quartic factor
which is unacceptable, since it would lead to a = c = 0, and an equation of
26
degree six:
b6 (2α2 ζ −αβǫ−2αβγ +β 3 )+b5 (4α2 µ+2αβζ −αǫ2 −4αǫγ −4αγ 2 +β 2 ǫ+2β 2 γ)
+5b4 (2αβµ−αǫζ−2αγζ+β 2 ζ)+10b3 (−αζ 2 +β 2 µ)+5b2 (−2αµζ+βǫµ+2βγµ−βζ 2 )
+b(−4αµ2 −2βµζ +ǫ2 µ+4ǫγµ−ǫζ 2 +4γ 2 µ−2γζ 2 )−2βµ2 +ǫµζ +2γµζ −ζ 3 = 0
(4.11)
We can now, by a simple division, take α = µ = 1. (Here we are treating the
generic, αµ 6= 0, case). Thus the biquadratic relation (3.1) is reduced to:
X 2 Y 2 + γ˜ (X 2 + Y 2 ) + ǫ˜XY + 1 = 0
(4.12)
(where we indicated by a tilde the parameters of the equation resulting from
the homographic transformation.). The parametrisation of (4.12) can be
given in terms of elliptic functions We introduce the ansatz X = A sn(z),
Y = A sn(z + q) where sn(z) denotes an elliptic sine of argument z and
modulus k. Substituting this ansatz into the (4.12) we find A2 = k and
moreover k satisfies the second-degree equation:
k 2 + γ˜ +
1
ǫ˜2 −
k+1=0
γ˜ 4˜
γ
(4.13)
Having obtained k from this equation we can compute q through k sn2 (q) +
1 = 0. Thus the biquadratic relation (4.8) can indeed be parametrised in
terms of elliptic functions.
We turn now to the asymmetric case. The invariant, with the same
conventions as for (4.8), is now:
αx2 y 2 + βx2 y + γx2 + δxy 2 + ǫxy + ζx + κy 2 + λy + µ = 0
(4.14)
We introduce two distinct homographic transformations x = (aX + b)/(cX +
d) and y = (eY + f )/(gY + h). As in the symmetric case we can take
d = h = 1 and, in order to better organise the calculations we put a = sc,
e = tg. We choose the parameters c and g so as to put κ
˜ = γ˜ and α
˜ =µ
˜.
We find the relations:
αb2 t2 + βb2 t + γb2 + δbt2 + ǫbt + ζb + κt2 + λt + µ
c2
=
g2
αf 2 s2 + βf s2 + γs2 + δf 2 s + ǫf s + ζs + κf 2 + λf + µ
27
(4.15)
c2 g 2 =
αb2 f 2 + βb2 f + γb2 + δbf 2 + ǫbf + ζb + κf 2 + λf + µ
αt2 s2 + βts2 + γs2 + δt2 s + ǫts + ζs + κt2 + λt + µ
(4.16)
The parameters b, f, s, t are chosen so as to put to zero the coefficients
˜ δ,
˜ ζ,
˜ λ.
˜ We find
β,
δbf 2 + ǫbf + ζb + 2κf 2 + 2λf + 2µ
s=−
2αbf 2 + 2βbf + 2γb + δf 2 + ǫf + ζ
(4.17)
βb2 f + 2γb2 + ǫbf + 2ζb + λf + 2µ
2αb2 f + βb2 + 2δbf + ǫb + 2κf + λ
(4.18)
t=−
There remain two equations for f and b. Taking the resultant for f , say,
we obtain for b an equation of degree 20. However it turns out that the
polynomial of degree 20 factorizes into two quartic ones and the square of a
polynomial of degree 6. These expressions, obtained with the help of computer algebra, are prohibitively long for a display here. Still, we were able
to show that the roots of the two quartic polynomials were unacceptable:
they lead to the vanishing or divergence of c and g, in which cases the whole
calculation collapses. Thus, as in the symmetric case, the condition is given
in the form of an equation of degree 6 (for which, generically, no problem
arises). Once b is obtained, f can be computed from the solution of a quartic
equation.
Thus in the end, after all the simplifications have been implemented,
(4.14) is reduced to precisely (4.12). Thus the solution of the full “asymmetric” biquadratic relation is again given in terms of elliptic functions. However,
since the two homographic transformations which take us back from the elliptic sines to the x, y that parametrise (4.14) are not the same for x and y,
the solutions in the asymmetric case are not simply related as in the symmetric case where one is the ‘upshift’ of the other. Note however, that only
the homography is different for x and y. The step q of the argument of the
elliptic function is the same at each iteration.
4.1
The A1 matrices for various QRT mappings
In this section we are going to show some examples of mappings which can be
analysed with the tools developed above. It turns out that one can choose
the A1 matrix to depend only on the ‘family’ of the equation and put all
the details into the A0 matrix. For a given equation, once the A1 matrix is
known, the construction of the corresponding A0 is elementary. The utility of
28
the mappings we give below resides alos in the fact that they are autonomous
form of some discrete Painleve transcendents. In what follows we present the
results without their derivation: once the form of the matrix is given one can
verify the results in a straightforward way. We give the general form of the
equation and the corresponding A1 matrix


0 0 0
(I)
xn+1 + xn−1 = f (xn )
A1 = 0 0 0
0 0 1
(II)
(III)
(IV)
(V)
(VI)
(VII)
(VIII)

0 0 0
A1 = 0 1 0
0 0 0

xn+1 xn−1 = f (xn )
(xn+1 + xn )(xn + xn−1 ) = f (xn )
(xn+1 xn − 1)(xn xn−1 − 1) = f (xn )
(xn+1 +xn +2z)(xn +xn−1 +2z)
(xn+1 +xn )(xn +xn−1 )
= f (xn )


0 0 0
A1 = 0 0 1
0 1 0


0 0 0
A1 = 0 1 0 
0 0 −1


0 0 1
A1 = 0 2 2z 
1 2z 0


1
0
0
(xn+1 xn −z 2 )(xn xn−1 −z 2 )
= f (xn )
A1 = 0 −z 2 − 1 0 
(xn+1 xn −1)(xn xn−1 −1)
0
0
z2


0
0
1
(xn+1 −xn −z 2 )(xn−1 −xn −z 2 )+xn z 2
= f (xn )
A1 = 0 −2 −2z 2 
xn+1 −2xn +xn−1 −2z 2
1 −2z 2
z4


0
0
z4
(xn+1 z 2 −xn )(xn−1 z 2 −xn )−(z 4 −1)2

0
= f (xn ) A1 =  0 −z 2 (z 4 + 1)
(xn+1 z −2 −xn )(xn−1 z −2 −xn )−(z −4 −1)2
4
4
2
z
0
(z − 1)
29
The forms presented above correspond to symmetric mappings but they can
be extended to asymmetric ones directly, the A1 matrix being the same. To
these cases one must add the explicitly asymmetric one


0 0 0
(IX)
xn+1 + xn = f (yn ), yn yn−1 = g(xn )
A1 = 0 0 1
0 0 0
Now once the A0 , A1 matrices are obtained, one can proceed to the explicit integration of the mapping. To do this, one uses the invariant and the
initial conditions in order to compute the integration constant K. Then one
constructs the corresponding α, β... of (4.8), (4.14) through α = α0 + Kα1 ,
β = β0 + Kβ1 etc. The important remark is that these equations depend explicitly on the integration constant. Thus the homographic transformations
and the details of the elliptic functions (modulus k and step q) are different
for every initial condition. This explains why the brute force computations
of the solutions of a given QRT mapping are particularly hard.
Remark 4.1. In 2004 Tsuda [41] somehow solved the miracle of integrability
of the QRT mapping. Practically he proved that the general QRT mapping is
nothing but the famous group law on an elliptic curve. More precisely if P1 +
P2 = P3 is the addition on the elliptic curve in the group structure then one
can coordonatize the points Pi through P1 = (xn , yn−1 ), P2 = (xn+1 , y), P3 =
(xn+1 , yn+1 ) from (2.3a) and (2.3b)
30
5
Rational surfaces and elliptic fibrations
Starting from this chapter we are goingt to study the singularity confinement in a rigurous way. Practically we shall show that a rigurous singularity
analysis can be used to integrate effectively the mapping (in this case by constructing the invariants). In this spirit we are going to use some elementary
tools from algebraic geometry o rational elliptic surfaces and a s a first outcome both singularity confinement and algebraic entropy will aquire rigurous
formulation. In addition we shall see how singularity analysis will give also
symmetries and the method of linearisation for linearisable systems (here
the problem is more complicated since in the case of linearisable systems the
number of singularities is infinite). Various examples will be given and in the
end we shall focus on tropical (ultradiscrete) mappings.
5.1
Discrete mappings and surfaces
In order to see how we go from mappings to surfaces we start from the same
example (3.15)
xn+1 + xn−1 + xn = a/xn
It is an order two equation which can be written as a system defined on C2
(or P2 if we include infinities):
xn+1 = yn
φ:
.
(5.1)
yn+1 = −xn − yn + yan
It can be seen also as a chain of birational mappings ... → (x, y) →
(x, y) → (¯
x, y¯) → ... where x = xn−1 , x = xn , x¯ = xn+1 and so on.
Each step is an automorphism of the field of rational functions C(x, y).
Now singularity confinement means:
(f, 0) → (0, ∞) → (∞, ∞) → (∞, 0) → (0, f )
| {z }
| {z }
| {z }
| {z }
| {z }
(x0 ,y0 )
(x1 ,y1 )
(x2 ,y2 )
(x3 ,y3 )
(x4 ,y4 )
The secret is the follwing: if (x0 , y0 ) = (f, ǫ) then the foolowing products are
finite
x2
x1 y1 = a + O(ǫ),
= −1 + O(ǫ), x3 y3 = −a + O(ǫ)
y2
31
Now let us construct a surface by glueing
1
1
2
2
∪ x1 y1 ,
C ∪ C = x1 ,
x1 y1
y1
But this is nothing but blow up of the affine space Spec C[x, Y ] with the
center (x, Y ) = (0, 0) which gives the surface (Y = 1/y):
X1 = {(x, Y, [z0 : z1 ]) ∈ Spec C[x, Y ] × P1 |xz0 = Y z1 } =
2
= Spec C[x, 1/xy] ∪ Spec C[xy, 1/y]
So by blowing up C in the points (x1 , y1 ) = (0, ∞), (x2 , y2 ) = (∞, ∞), (x3 , y3 ) =
(∞, 0) the equation then make sense on this new surface given by glueing such
affine schemes. Accordingly we do analize any discrete order two nonlinear
equation by identifying the singularities and blow them up.
5.2
Preliminaries on rational elliptic surfaces
We begin by the following definition: A complex surface X is called a rational
elliptic surface if there exists a fibration given by the morphism: π : X → P1
such that:
• for all but finitely many points k ∈ P1 the fibre π −1 (k) is an elliptic
curve
• π is not birational to the projection : E × P1 → P1 for any curve E
• no fibers contains exceptional curves of first kind.
Blowing up: Let X be a smooth projective surface and let p be a point on X.
There exist a smooth projective surface X ′ and a morphism π : X ′ → X
such that π −1 (p) ∼
= P1 and π represents a biholomorphic mapping from
X ′ − π −1 (p) → X − (p). The morphism is called blow-down and the correspondence π −1 is called blow-up of X at p as a rational mapping. For
example if X is the space C2 and p is a point of coordinate (x0 , y0 ) then we
denote blow-up of X in p
X ′ = {(x − x0 , y − y0 ; ζ0 : ζ1 ) ∈ C2 × P1 |(x − x0 )ζ0 = (y − y0)ζ1 }
by (we use the coordinates notation rather than glueing affine schemes)
π : (x, y) ←− (x − x0 , (y − y0 )/(x − x0 )) ∪ ((x − x0 )/(y − y0 ), y − y0 )
32
Space of initial conditions: Let Yi be smooth projective surfaces and let
{φi : Yi → Yi+1 } be a sequence of dominant rational mappings. A sequence of
rational surfaces {Xi } is called the space of initial conditions for the sequence
φi if each φi is lifted by blowing ups to the mappings φ′i : Xi → Xi+1 such
that the set of indeterminate points of φ′i is empty.
Also we denote the group of divizors of a variety X by Div(X). The
Picard group of X is the group of isomorphism classes of invertible sheaves
on X and it is isomorphic to the group of linear equivalence classes of divisors
on X. We denote it by Pic(X).
Total transform and proper transform: Let π −1 : X → Y be the blow up
at the point p and D be a divisor on X. The bundle mapping π ∗ (D) on Y
is called total transform of D and for any analytic subvariety V on X the
closure of π −1 (V − p) in Y is called the proper tranform of V.
Let X be a surface obtained by N times blowing up of P1 × P1 . Then the
Picard group Pic(X) is isomorphic to a Z module (the Neron-Severi lattice)
with the form:
N
X
Z Hx + Z Hy +
Z Ei
i=1
where Hx , Hy are the proper transforms of lines x = const., y = const. and
Ei is the total transform of the i − th blow up. In addition the intersection
numbers of two divisors on X are given by the following basic formulas (valid
for any i, j = 1...N ):
Hx · Hy = 1, Ei · Ei = −1, Ei · Ej = Ei · Hx = Ei · Hy = Hx · Hx = Hy · Hy = 0
A rational surface X is called a generalized Halphen surface if the anticanonical divisor
Pclass −KX is uniquely decomposed into effective divisors as
[−KX ] = D =
mi Di (mi ≥ 1) such that Di · KX = 0 Generalized Halphen
surfaces can be obtained from P2 by succesive 9 blow-ups. They can be classified by D in elliptic, multiplicative and additive type. A rational surface X
is called a Halphen surface of index m if the dimension of the linear system
| − kKX | = 0, k = 1, m − 1 and | − kKX | = 1, k = m. A Halphen surface
of index m is also referred to be a rational elliptic surface of index m. The
linear system | − kKX | is the set of curves in P2 (resp. P1 × P1 ) of degree
3k (resp. 4k) passing through each point of blow-up with multiplicity k.
It is known that any Halphen surface of index m contains a unique cubic
curve with multiplicity m It is known that if m ≥ 2 a Halphen pencil of
index m contains a unique cubic curve C with multiplicity m, i.e. C is the
33
unique element of | − KX |. It is well known that if X is a Halphen surface
of index m and C is nonsingular, then k(P1 + · · · + P9 − 3P0 ) is not zero for
k = 1, . . . , m−1 and zero for k = m (here + is the group law on C, P1 , . . . , P9
are base points of blow-ups and 3P0 is equal by the group law to 3 crossing
points with a generic line in P2 ). Conversely, for a nonsingular cubic curve
C in P2 , if k(P1 + · · · + P9 − 3P0 ) is not zero for k = 1, . . . , m − 1 and zero
for k = m, then there exists a family of curves of degree 3m passing through
P1 , . . . , P9 with multiplicity m, which constitutes a Halphen pencil of index
m(see chap. 5 §6 of [62] for more details).
It is known that a rational elliptic surface can be obtained by 9 blow-ups
from P2 and that the generic fiber of X can be put into a Weierstrass form:
f (x, y, k) = y 2 + a1 xy + a3 y − x3 − a2 x2 − a4 x − a6 ,
where all the coefficients ai depend on k. Singular fibers can be computed
easily by the vanishing of the discriminant:
∆ ≡ −b22 b8 − 8b34 − 27b26 + 9b2 b4 b6 ,
where b2 = a21 +4a2 , b4 = 2a4 +a1 a3 , b6 = a23 +4a6 , b8 = a21 a6 +4a2 a6 −a1 a3 a4 +
a2 a23 − a24 . The discriminant has degree 12 which gives the number of singular
fibers together with their multiplicities. The singularities have been classified
by Kodaira according to the irreducible components of singular fibers.
Now for any nonlinear birational discrete equation of the form:
xn+1 = f (xn , yn )
yn+1 = g(xn , yn )
In [57], Sakai showed that every discrete Painlv´e equation can be obtained
as a translational component of an affine Weyl group which acts on a family
of generalized Halphen surfaces, i.e. a rational surface with special divisors
obtained by 9-blow-ups from P2 . From this viewpoint the Quispel-RobertsThomson (QRT) mappings [33] are obtained by specializations of the surfaces
so that they admit elliptic fibrations.
In autonomous setting, Diller and Favre [63] showed that if a K¨ahler
surface S admits an automorphism ϕ of infinite order, then (i) ϕ is ”linearizable”, i.e. it preserves the fibrations of a ruled surface [72]; (ii) ϕ preserves
an elliptic fibration of S; or (iii) the algebraic (or topological) entropy of ϕ is
34
positive. The typical example of the second case is so called the QRT mappings [33], while mappings not belonging to the QRT family are discovered
by several authors [70, 73, 74].
In this part, we classify these types of mappings by their relation with
rational elliptic surfaces. For this purpose, we consider not only rational
elliptic surfaces but also generalized Halphen surfaces. In next section, we
propose a classification of autonomous rational mappings preserving elliptic
fibrations. We also show an equivalent condition when a generalized Halphen
surface becomes a Halphen surface of index m. Although our classification
is rather simple, existence of (simple) examples is nontrivial. In the next
section, we extend this result into the case where C is singular by using ”the
period map” for generalized Halphen surfaces.
5.3
Classification
Let X be a rational elliptic surface obtained by 9 blow-ups from P2 . The
main result is the following classification.
Classification Let m be a positive integer, ϕ an automorphism of X which
preserves the elliptic fibration αf0 (x, y, z) + βg0 (x, y, z) = 0. Such cases are
classified as follows.
i-m) ϕ preserves α : β and the degree of fibers is 3m;
ii-m) ϕ does not preserve α : β and the degree of fibers is 3m.
Remark 5.1.
• The QRT mappings belong to Case i-1) [41].
• In case ii-m), elliptic fibrations admit exchange of fibers.
• The integer m corresponds to the index m of X as a Halphen surface.
• It is well known (for example van Hoeji’s gave an algorithm [69] and [60]
used it) that there exists a birational transformation on P2 which maps
an (possibly singular) elliptic curve in P2 , αf0 (x, y, z)+βg0 (x, y, z) = 0,
into the Wierstrass normal form. Since in general the coefficients
of this transformation are algebraic on a rational function α/β =
g0 (x, y, z)/f0 (x, y, z), there exists a bialgebraic transformation from a
Halphen surface of index m to that of index one that preserves the
35
elliptic fibrations. On the other hand, Proposition 11.9.1 of [64] shows
non-existence of a birational transformation from the Halphen surface
of index m to that of index m′ (m 6= m′ ) that preserves the elliptic
fibrations. (Precisely saying, Proposition 11.9.1 of [64] claims that if
an example of a mapping of the case i-2 is infinite order, then it is not
birationally conjugate to a mapping of the class i-1. But its proof is
still effective for the above assertion.) If two infinite order mappings
preserving rational elliptic fibrations are conjugate with each other by
a birational mapping ψ, the mapping ψ preserves the elliptic fibrations.
Thus, two infinite order mappings belonging to different classes of the
above classification are not birationally conjugate with each other.
In the rest of this section, we characterize Halphen surfaces as generalized
Halphen surfaces.
Let X be a generalized Halphen surface and Q the root lattice defined as
the orthogonal complement of D with respect to the intersection form
P and
ω a meromorphic 2-form on X with Div(ω) = −Dred , where Dred = si Di .
Then, the 2-form ω determines the period mapping χ from Q to C by
Z
χ(α) = ω
α
P
in modulo γ Z χ(γ), where the summation is taken for all the cycles on
Dred (see examples in the next section and [57] for more details). Note that
if X is not a Halphen surface of index one, then the divisor D and thus ω
(modulo a nonzero constant factor) are unique. The divisor D (or X itself
if X is not a Halphen surface of index one) is called elliptic, multiplicative,
or additive type if the rank of the first homology group of Dred is 2, 1, or 0
respectively.
Theorem 5.2.
(ell) If a member of | − KX | is of elliptic type, then X is a Halphen pencil of
index m iff χ(−kKX ) 6= 0 for k = 1, . . . , m − 1 and χ(−mKX ) = 0.
(mult) If a member of |−KX | is of multiplicative type, then the same assertion
holds as in the elliptic case.
(add) If a member of | − KX | is of additive type, then X is a Halphen pencil
of index 1 iff χ(−KX ) = 0, and never a Halphen pencil of index m ≥ 2.
Proof. Case (ell) is a classical result (see Remark 5.6.1 in [62] or references
therein). Case (mult) and case (add) of index 1 are Proposition 23 in [57].
36
Similar to that proof, we can vary D and χ continuously to nonsingular case.
Indeed, let P1 , . . . , P9 be the points of blow-ups (possibly infinitely near, we
assume P9 is the point for the last blow-up) and f0 be the cubic polynomial
defining D. There exists a pencil of cubic curves Cλ : fλ = f0 + λf1 = 0
λ ∈ P1 passing through the 8 points P1 , . . . , P8 . For small λ, the cubic curve
Cλ is close to D, and the meromorphic 2-form ωλ for Cλ is also close to ω.
Let P9′ be a point close to P9 on Cλ such that
Z
Z
′
ω =
lim χλ (−mKX ′ ) = lim
ω ′ = χ(−mKX )
λ→0
λ→0
−mKX ′
−mKX
holds,y where X ′ is the surface obtained by blow-ups at P1 , . . . , P8 and P9′
instead of P9 . Thus, χλ (−mKX ′ ) 6= 0 holds if χ(−mKX ) 6= 0 for small
λ, and therefore X does not have a pencil of degree 3m. Conversely, if
χ(−mKX ) = 0, then χ′ (−mKX ′ ) is close to zero, and there exists P9′′ close
to P9′ on C ′ such that χλ (−mKX ′′ ) = 0. Thus, we have
lim χλ (−mKX ′′ ) = χ(−mKX ).
λ→0
Since X ′′ has (at least) a pencil of curves of degree 3m passing through
the 9 points with multiplicity m and this condition is closed in the space of
coefficients of polynomials defining curves, X also has the same property.
Remark 5.3. In Painlev´e context, for multiplicative case, χ is normalized
so that χ(γ) = 2πi for a simply connected cycle γ on some Di , and the
parameter “q” is defined as q = exp χ(−KX ), i.e. the condition χ(−mKX ) =
0 corresponds to q m = 1. We must point out here that in [67] and [68] similar
study has been done
√ on q-Painlev´e equations, and it is reported that Eq. (3.1)
of [67] with q = −1 preserves degree (4,4) pencil, which seems contradict
to the above theorem, but there the definition of q is different from ours (its
square root is our q).
6
Examples
In this section, we are going to give examples for case i-2, ii-1 and ii-2. A
typical example of Case i-1 is the QRT mappings. There is some literature
on their relation to rational elliptic surfaces [41, 64], and we are not going to
discuss it here. In the first subsection, we investigate the action on the space
37
of initial conditions of some mapping of Case ii-1, which was proposed in [74].
In the second subsection, we show that one of the HKY mappings belongs
to Case i-2. Theoretically, from Theorem 5.2, we can construct mappings of
the type i-m for any integers. Actually, let φ(q) be some q-discrete Painlev´e
equation and q a primitive m-th root of unity, then φm (q) is autonomous
and preserves the Halphen fibration of index m. However, the degrees of
mappings obtained in this way are very high. The HKY mapping is much
simpler example. In the third subsection, we construct some example for
Case ii-2, which we believe as the first example for this case.
6.1
Case ii-1
We start with a mapping [7, 74, 67] which preserves elliptic fibration of degree
(2, 2) but exchanges the fibers:
xn+1 = −xn−1
(xn − a)(xn − 1/a)
.
(xn + a)(xn + 1/a)
(6.1)
In this subsection, studying space of initial conditions (values), we compute
the conserved quantity, the parameter “q” and all singular fibers. We also
(1)
clarify the relation with the q-discrete Painlev´e VI equation (qP (A3 ) in
(1)
Sakai’s notation) by deautonomizing the mapping (6.1), where the label A3
corresponds to the type of space of initial conditions.
First of all, in order to compactify the space of dependent variables, we
write the equations in projective space as a two component system:
φ : P1 × P1 → P1 × P1 , φ(x, y) = (x, y),
x=y
y = −x
(y − a)(y − 1/a)
.
(y + a)(y + 1/a)
(6.2)
We use P1 × P1 instead of P2 just because the parameters of blowing-up
points become easy to write. The projective space P1 × P1 is generated by
the following coordinate system (X = 1/x, Y = 1/y):
P1 × P1 = (x, y) ∪ (X, y) ∪ (x, Y ) ∪ (X, Y ).
38
The indeterminate points for the mappings φ and φ−1 are
P1 : (x, y) = (0, −a), P2 : (x, y) = (0, −1/a),
P3 : (X, y) = (0, a), P4 : (X, y) = (0, 1/a),
P5 : (x, y) = (a, 0), P6 : (x, y) = (1/a, 0),
P7 : (x, Y ) = (−a, 0), P8 : (x, Y ) = (−1/a, 0).
Let X be the surface obtained by blowing up these points. Then, φ is lifted
to an automorphism of X. Such a surface X is called the space of initial
conditions. More generally, if a sequence of mappings {φn } is lifted to a
sequence of isomorphisms from a surface Xn to a surface Xn+1 , each surface
Xn is called the space of initial conditions.
The Picard group of X is a Z-module:
Pic(X) = Z Hx ⊕ Z Hy ⊕
8
M
Z Ei ,
i=1
where Hx , Hy are the total transforms of the lines x = const., y = const. and
Ei are the total transforms of the eight points of blow-ups. The intersection
form of divisors is given by Hz · Hw = 1 − δzw , Ei · Ej = −δij , Hz · Ek = 0
for z, w = x, y. Also the anti-canonical divisor of X is
−KX = 2Hx + 2Hy −
8
X
Ei .
i=1
P8 Let us denote an element of the Picard lattices by A = h0 Hx + h1 Hy +
i=1 ei Ei (hi , ej ∈ Z), then the induced bundle mapping is acting on it as
φ∗ (h0 , h1 , e1 , ..., e8 )








=(h0 , h1 , e1 , ..., e8 ) 







2
1
1
1
1
1
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
39
0
0
0
0
0
0
0
0
1
0

0 −1 −1 −1 −1
0 0
0
0
0 

0 0
0 −1 0 

0 0
0
0 −1 

0 −1 0
0
0 
.
0 0 −1 0
0 

0 0
0
0
0 

0 0
0
0
0 

0 0
0
0
0 
1 0
0
0
0
It preserves the decomposition of −KX =
P3
i=0
Di :
D0 = H x − E 1 − E 2 , D1 = H y − E 5 − E 6
D2 = H x − E 3 − E 4 , D3 = H y − E 7 − E 8 ,
(6.3)
(1)
which constitute the A3 type singular fiber: xy = 1.
One can see that the elliptic curves
F ≡αxy − β((x2 + 1)(y 2 + 1) + (a + 1/a)(y − x)(xy + 1)) = 0
⇔ kxy − ((x2 + 1)(y 2 + 1) + (a + 1/a)(y − x)(xy + 1)) = 0
correspond to the anti-canonical class (these curves pass through all Ei ’s for
any α : β). This family of curves defines a rational elliptic surface. One can
see that even though the anti-canonical class is preserved by the mapping,
the each fiber is not. More precisely, the action changes k in −k.
So, as a conclusion, the dimension of the linear system corresponding to
the anti-canonical divisor is 1. It can be written as αf1 (x, y) + βf2 (x, y) =
0 ⇔ kf1 (x, y)+f2 (x, y) = 0 for α : β ∈ P(C) and deg f = deg g = (2, 2). This
elliptic fibration is preserved by the action of the dynamical system but not
trivially in the sense that the fibers are exchanged. The conserved quantity
becomes higher degree as (f /g)ν for some ν > 1. In our case ν = 2 and the
invariant is exactly the same as the result of [74].
Remark 6.1. In order to have a Weierstrass model, we perform some homographic transformations according to the algorithm of Schwartz [71]. Then,
after long but straightforward calculations, we can compute the roots of the
elliptic discriminant ∆(k) as
k1 = 0,
k2,3 = ±4(1 + a2 )/a,
k4,5 = ±(1 − a2 )2 /a2 ,
k6 = ∞,
(1)
multiplicity = 2
multiplicity = 1
multiplicity = 2
multiplicity = 4.
(1)
(1)
We have A1 singular fiber for k1 and k4,5 , A0 fiber for k2,3 , and A3 fiber
for k6 . The mapping acts on these singular fibers as an exchange as (k1 →
k1 , k2 → k3 → k2 , k4 → k5 → k4 , k6 → k6 ).
Remark 6.2. If a surface is a generalized Halphen surface but not a Halphen
surface of index 1, then the anti-canonical divisorP−KX is uniquely decomposed to a sum of effective divisors as −KX =
mi Di and we can characterize the surface by the type the decomposition. However, if the surface
40
is Halphen of index 1, it may have several types of singular fibers as this
example.
Next, we consider deautonomization of the mapping φ. For that, we will
use the decomposition (6.3) of −KX which is preserved by the mapping,
though the decomposition and hence the deautonomization are not unique
(the fiber corresponding to k = 0 is also preserved).
The affine Weyl group symmetries are related to the orthogonal complement of Dred = {D1 , . . . , D4 }. In order to see this, we note that rank Pic(X) =
rank hHx , Hy , E1 , ...E8 iZ = 10. The orthogonal complement of Dred :
hDi⊥ = {α ∈ Pic(X)|α · Di = 0, i = 0, 3}
has 6-generators:
hDi⊥ = hα0 , α1 , ..., α5 iZ
α0 = E 1 − E 2 , α1 = E 3 − E 4 , α2 = Hy − E 1 − E 3
α3 = Hx − E 5 − E 7 , α4 = E 5 − E 6 , α5 = E 7 − E 8 .
Figure 1: Singular fiber and orthogonal complement.
Related to them, we define elementary reflections:
wi : Pic(x) → Pic(X), wi (αj ) = αj − cij αi ,
41
where cji = 2(αj · αi )/(αi · αi ). One can easily see that cij is a Cartan
L
(1)
matrix of D5 -type for the root lattice Q = 5i=0 Z αi . We also introduce
permutations of roots:
σ10 : (α0 , α1 , α2 , α3 , α4 , α5 ) 7→ (α1 , α0 , α2 , α3 , α4 , α5 )
σtot : (α0 , α1 , α2 , α3 , α4 , α5 ) 7→ (α5 , α4 , α3 , α2 , α1 , α0 ).
The group generated by reflections and permutations becomes an extended
affine Weyl group:
f(D5(1) ) = hw0 , w1 , ..., w5 , σ10 , σtot i.
W
This extended affine Weyl group can be realized as an automorphisms of
a family of generalized Halphen surfaces which are obtained by allowing the
points of blow-ups to move so that they preserve the decomposition of −KX
as
P1 : (x, y) = (0, a1 ),
P3 : (X, y) = (0, a3 ),
P5 : (x, y) = (a5 , 0),
P7 : (x, Y ) = (a7 , 0),
P2 : (x, y) = (0, a2 ),
P4 : (X, y) = (0, a4 ),
P6 : (x, y) = (a6 , 0),
P8 : (x, Y ) = (a8 , 0),
which can be normalized as a1 a2 a3 a4 = a5 a6 a7 a8 = 1. Accordingly, our
f (D5(1) ) and deautonomized
mapping lives in an extended affine Weyl group W
as

 x = a1 a2 y
(y − a3 )(y − a4 )
φ˜ :
 y = −x
(y − a1 )(y − a2 )
with
(a1 , a2 , a3 , a4 , a5 , a6 , a7 , a8 , q)
√
√
q
q
1
1
7→ (− √ , − √ , −
,−
, a3 , a4 , a1 , a2 , q),
qa6
qa5
a8
a7
where
q=
a1 a2 a7 a8
.
a3 a4 a5 a6
42
(6.4)
This mapping can be decomposed by elementary reflections as
φ˜∗ =σ10 ◦ σtot ◦ σ10 ◦ σtot ◦ w2 ◦ w1 ◦ w0 ◦ w2 ◦ w1 ◦ w0
and acts on the root lattice as
(α0 , α1 , α2 , α3 , α4 , α5 )
7→ (−α5 , −α4 , −α3 , α2 + 2α3 + α4 + α5 , α0 , α1 ).
Hence, φ˜4 is a translational element of the extended affine Weyl group, and
(1)
therefore one of the q-Painlev´e VI equations (qP (A3 )) in Sakai’s sense, while
the original q-Painlev´e VI studied in [57] was

y

 x =−
a1 a2
qPVI :
(y − a1 )(y − a2 )

 y =−
x(y − a3 )(y − a4 )
with
(a1 , a2 , a3 , a4 , a5 , a6 , a7 , a8 , q)
a8
a7
√
√
7→ (− qa5 , − qa6 , − √ , − √ , −a1 , −a2 , −a3 , −a4 , q),
q
q
which is decomposed by elementary reflections as
qPVI = σ10 ◦ w1 ◦ w0 ◦ w2 ◦ w1 ◦ w0 ◦ w2 ◦ w1 ◦ w0
and acts on the root lattice as
(α0 , α1 , α2 , α3 , α4 , α5 )
7→ (−α4 , −α5 , −α3 , −α2 + δ, −α0 , −α1 )
(δ = α0 + α1 + 2α2 + 2α3 + α4 + α5 ).
At the last of this subsection, we define the period map χ : Q → C and
compute q (6.4) by using
ω=
1 dx ∧ dy
.
2πi xy
43
(6.5)
For example, χ(α0 ) is computed as follows. The exceptional divisors E1 and
E2 intersect with D0 at (x, y) = (0, a1 ) and (0, a2 ), and χ(α0 ) is computed as
Z
1 dx ∧ dy
χ(α0 ) =
xy
|x|=ε, y=a2 ∼a1 2πi
Z a1
dy
=−
y
a2
a2
= log ,
a1
where y = a2 ∼ a1 denotes a path from y = a2 to y = a1 in D1 . According
to the ambiguity of paths, the result should be considered in modulo 2πi Z.
Similarly, we obtain
a2
a3
a1
χ(α0 ) = log , χ(α1 ) = log , χ(α2 ) = log ,
a1
a4
a3
a7
a5
a8
χ(α3 ) = log , χ(α4 ) = log , χ(α5 ) = log ,
a5
a6
a7
and therefore we have
χ(−KX ) = log
a1 a2 a7 a8
a3 a4 a5 a6
and q as (6.4) (see Remark 5.3). For the mapping φ, we have q = 1.
6.2
Case i-2
We consider the following HKY mapping which is a symmetric reduction of
qPV for q = −1 [70] (also [66] pg. 311).
(x − t)(x + t)
y(x − 1)
y¯ = x
x¯ =
(6.6)
We define the space of initial conditions as a rational surface obtained by
blow-ups from P1 × P1 at 8 points:
P1 : (x, y) = (a1 , 0) = (t, 0),
P3 : (x, y) = (0, a3 ) = (0, t),
P5 : (x, y) = (1, ∞),
P7 : (x, y) = (∞, ∞),
P2
P4
P6
P8
44
: (x, y) = (a2 , 0) = (−t, 0)
: (x, y) = (0, a4 ) = (0, −t)
: (x, y) = (∞, 1)
: (x, x/y) = (∞, a5 ) = (∞, 1),
where ai ’s wii be used for deautonomization later. The system acts the
surface as a holomorphic automorphism.
Again we investigate the linear system of the anti-canonical divisor class
−KX = 2Hx + 2Hy − E1 − · · · − E8 . For the example, dim −KX is zero and
dim −2KX is one. Actually, we have
| − 2KX | =αx2 y 2 + β(2x2 y 3 + 2x3 y 2 + x2 y 4 + x4 y 2 − 2x3 y 3 −
2xy 4 − 2x4 y + x4 + y 4 + 2t2 (xy 2 + x2 y − y 2 − x2 ) + t4 ) ≡ αf + βg,
and
g 2 3
k = = 2x y + 2x3 y 2 + x2 y 4 + x4 y 2 − 2x3 y 3 − 2xy 4 − 2x4 y
f
.
4
4
2
2
2
2
2
4
+ x + y + 2t (xy + x y − y − x ) + t ) (x2 y 2 )
is the conserved quantity. So it belongs to Case ii-1.
Remark 6.3. We say a curve f (x, y) = 0 passes through a point (x0 , y0 ) with
j f (x ,y )
0 0
= 0 for any j ≤ m and p + q = j. The calculation
multiplicity m if ∂ ∂x
p ∂y q
of multiplicity at P8 is very sensitive. For example, for f (x, y) = x2 y 2 , we
relate F (X, Y ) = X 2 Y 2 so that the sum of degrees is (4, 4). Since this curve
passes through P7 with multiplicity 2, the proper transform of the curve in
the coordinate: (u, v) = (X, Y /X) is given by F (u, uv)/u2 = u2 v 2 , which
passes through P8 : (u, v) = (0, 1) with multiplicity 2.
RemarkP6.4. The unique anti-canonical divisor −KX is decomposed as
−KX = 4i=0 Di by
D0 = H y − E 1 − E 2 , D1 = H x − E 6 − E 7
D2 = E 7 − E 8 , D3 = H y − E 5 − E 6 , D4 = H x − E 3 − E 4 ,
(1)
which constitute A4 -type singular fiber xy = 1. The orthogonal complement
of Di ’s is generated by
α0 = Hx + Hy − E 1 − E 3 − E 7 − E 8 , α1 = E 1 − E 2
α2 = Hx − E 1 − E 5 , α3 = Hy − E 3 − E 6 , α4 = E 3 − E 4 ,
45
(1)
which forms the Dynkin diagram of typeL
D4 . Let ω be the same as (6.5),
then the period map χ : Q → C for Q = 4i=0 Z αi is computed as
a1
1
a3
, χ(α1 ) = log , χ(α2 ) = log ,
χ(α0 ) = log −
a1 a5
a2
a1
a4
χ(α3 ) = log a3 , χ(α4 ) = log ,
a3
and therefore
χ(−KX ) = log
a3 a4
+ πi
a1 a2 a5
Hence we can take q as
a3 a4
a1 a2 a5
and we have q = −1 for the original mapping.
q=−
Figure 2: Singular fiber and orthogonal compliment.
(1)
The mapping (6.6) can be deautonomized to one of qPV (qP (A4 )) as
a5 (x − a1 )(x − a2 )
y(x − 1)
y =x
x =
with
(a1 , a2 , a3 , a4 , a5 , q) 7→ (
1
a4 a3
, , a1 , a2 , − , q),
q q
a5
46
which acts on the root lattice as
(α0 , α1 , α2 , α3 , α4 ) 7→ (α2 + α3 + α4 , −α4 , −α3 , α0 + α3 + α4 , α1 ).
While the original qPV mapping was
a5 (x − a1 )(x − a2 )
y(x − 1)
y =x
x =
(same with the above) with
(a1 , a2 , a3 , a4 , a5 , q) 7→ (
a4 a3
1
, , a2 , a1 , − , q),
q q
a5
which acts on the root lattice as
(α0 , α1 , α2 , α3 , α4 )
7→ (α1 + α2 + α3 + α4 , −α4 , −α3 , α0 + α1 + α3 + α4 , −α1 ).
6.3
Case ii-2
We consider the following mapping ϕ:

x(−ix(x + 1) + y(bx + 1))


 x¯ = y(x(x − b) + iby(x − 1))
ϕ:
,

x(x(x + 1) + iby(x − 1))

 y¯ =
b(x(x + 1) − iy(x − 1))
(1)
(6.7)
which is obtained by specializing one of qP (A5 ) equation. Notice that the
space of initial conditions for both qPIII and qPIV is the generalized Haphen
(1)
surface of type of A3 [57], and thus we may not be able to say that a
translational element of the corresponding affine Weyl group is one of qPIII
equations or qPIV equations.
The inverse of ϕ is

y(bxy − bx − by + 1)


 x = xy − x + by − 1
(6.8)
ϕ−1 :

−iy(bxy − bx − by + 1)(bxy + x − by + 1)

 y =
bx(xy − x − y − 1)(xy − x + by − 1)
47
and the space of initial conditions is obtained by blow-ups from P1 × P1 at 8
points:
P1 : (x, y) = (−1, 0),
P3 : (x, y) = (1, ∞),
P5 : (x, y) = (0, 0),
P7 : (x, y) = (∞, ∞),
P2
P4
P6
P8
: (x, y) = (0, 1/b)
: (x, y) = (∞, 1)
: (x, y/x) = (0, i)
: (x, x/y) = (∞, −ib).
Then ϕ acts the surface as a holomorphic automorphism.
For the above example, dim | − KX | is zero and dim | − 2KX | is one.
Actually, we have
| − 2KX | :
By ϕ, the parameter
0 = kf0 (x, y)− f1 (x, y)
= kx2 y 2 − ix(x + 1)2 − i(x + i)(x2 − 1)y .
+b(x − 1)2 y 2 − ix(y − 1) + y(by − 1)
k=
f1 (x, y)
f0 (x, y)
f1 (x, y)
f0 (x, y)
is mapped to −k. So,
2
k =
2
is the conserved quantity and ϕ belongs to Case ii-2.
We found this example by observing the following facts:
• Let X be a generalized Halphen surface of multiplicative type, then
exp(χ(−KX )) is the parameter q of the corresponding q-discrete Painlev´e
equation. From Theorem 5.2, if q is a primitive m-th root of unity, then
dim | − kKX | is 0 for k = 1, 2, . . . , m − 1 and 1 for k = m.
• Let ψ be an automorphism of the surface. If there exists another automorphism σ of the surface such that σ acts the base space of | − mKX |
nontrivially, then ϕ = σ ◦ ψ belongs to the case ii-2 unless it is finite
order.
48
(1)
First, we consider the family of generalized Halphen surfaces of type A5 .
Those surfaces are obtained by blow-ups from P1 × P1 at 8 points:
P1 : (x, y) = (b1 , 0),
P3 : (x, y) = (1, ∞),
P5 : (x, y) = (0, 0),
P7 : (x, y) = (∞, ∞),
P2
P4
P6
P8
: (x, y) = (0, 1/b2 )
: (x, y) = (∞, 1)
: (x, y/x) = (0, c)
: (x, x/y) = (∞, 1/(cb0 )).
The anti-canonical divisor xy = 0 is decomposed by
Hx − E 2 − E 5 , E 5 − E 6 , Hy − E 1 − E 5 ,
Hx − E 4 − E 7 , E 7 − E 8 , Hy − E 3 − E 7 ,
and their orthogonal complement is generated by
α0
α1
α2
β0
(β1
= Hx + Hy − E 5 − E 6 − E 7 − E 8
= Hx − E 1 − E 3
= Hy − E 2 − E 4
= Hx + Hy − E 1 − E 2 − E 7 − E 8
= Hx + Hy − E3 − E4 − E5 − E6 ).
The period map χ : Q → C for the same ω with (6.5) is computed as
χ(α0 ) = − log b0 , χ(α1 ) = − log b1 , χ(α2 ) = − log b2 ,
χ(β0 ) = − log(−cb0 b1 b2 ), χ(β1 ) = log(−c),
and therefore χ(−KX ) = − log(b0 b1 b2 ). We set q = (b0 b1 b2 )−1 .
The following actions generate the group of automorphisms of the family
49
Figure 3: Singular fiber and orthogonal complement.
(1)
(1)
of surfaces, whose type is A2 + A1 :
(x, y; b0 , b1 , b2 , c) is mapped to
1
x y(x − 1)
w α1 :
,
; b0 b1 , , b1 b2 , c
b1 x − b1
b1
b2 x(y − 1)
1
w α2 :
, b2 y; b0 b2 , b1 b2 , , c
b2 y − 1
b2
1 1 1 1
π : y, x;
, , ,
b0 b1 b2 c
1 cx
, ; b1 , b2 , b0 , c
ρ:
y y
cx(xy − x − y)
y(xy − x − y)
1
w β1 : −
,−
; b0 , b1 , b2 ,
xy + cx − y
(cxy − cx + y)
c
b1 1
1
σ:
,
,
; b0 , b1 , b2 ,
x b2 y
b0 b1 b2 c
wα0 = ρ−1 ◦ wα2 ◦ ρ and wβ0 = σ ◦ wβ1 ◦ σ.
Here, wαi acts as the elementary reflection of the affine Weyl group of
L
L
(1)
type
A
on
Z
α
and
trivially
on
i
2
i
j Z βj . Similarly, wβi acts trivially
L
on i Z αi and as the elementary reflection of the affine Weyl group of type
50
(1)
A1 on
L
j
Z βj . The generators (α0 , α1 , α2 , β0 , β1 ) are mapped by π, ρ, σ to
π :(α0 , α2 , α1 , β0 , β1 )
ρ :(α2 , α0 , α1 , β0 , β1 )
σ :(α0 , α1 , α2 , β1 , β0 ).
If q = (b0 b1 b2 )−1 = −1, then χ(−Kx ) = − log(−1) = −πi mod 2πi Z,
and | − 2KX |, i.e. the set of curves of degree (4, 4) passing through the
blow-up points with multiplicity 2, is given by
2 2
k0 x y + k1 c2 x4 (y − 1)2 + 2b1 cx2 (cxy − cx + y + b2 y 2 (xy − x − y))+
b21 (c2 x2 + 2cxy(b2 y − 1) + (y + b2 y 2 (x − 1))2 ) = 0.
Moreover, if c = i, then σ acts identically on the parameter space and maps
k1 /k0 to −k1 /k0 .
Let ψ = (wα1 ◦ wα2 ◦ ρ)2 , where wα1 ◦ wα2 ◦ ρ is the original qPIII equation
(x, y; b0 , b1 , b2 , c)
cx(b0 (cx − y) − y(b0 cx − y)) b0
cx − y
7→
,
;
, b1 q, b2 , c
b2 y(b0 cx − y) y((cx − y) − b2 y(b0 cx − y)) q
and acts on the root lattice as
(α0 , α1 , α2 , β0 , β1 )
7→ (α0 − δ, α1 + δ, α2 , β0 , β1 )
(δ = α0 + α1 + α2 = β0 + β1 ).
Then the mapping ψ acts trivially on the parameter space. Since it is very
intricate mapping, we restrict the parameters to b0 = 1/b, b1 = −1 and
b2 = b, then we have ϕ = σ ◦ ψ as (6.7), which acts on the root lattice as
(α0 , α1 , α2 , β0 , β1 )
7→ (α0 − 2δ, α1 + 2δ, α2 , β1 , β0 ).
As a conclusion, the mapping σ ◦ wα1 ◦ wα2 ◦ ρ ◦ wα1 ◦ wα2 ◦ ρ with the
(1)
full parameter b1 , b2 , b3 , c is one of qP (A5 ) equation and gives the mapping
by specializing of the parameters.
51
7
Q4 mapping
We said in the beginning of the previous chapter that Sakai showed every
discrete Painleve equation can be formulated as a translation in an affine
Weyl group which acts on a family of generalised Halphen surfaces obtained
by nine blow-ups from P2 (or eight blow ups from P1 × P1 ). This formulation
shows how to classify all discrete Painleve equations using this algebraic(1)
geometric framework. Naturally for the richest affine Weyl group (E8 ) the
(1)
translational component acting on a surface of type A0 represents a kind
of ”master” equation for all the other Painlev´e equations. This has been
obtained by Sakai under the name of elliptic Painlev´e equation and initially
had a very complicated form. Later on, Ohta, Ramani and Grammaticos [43]
found a regular form of an elliptic Painlev´e equation. We have to point out
that if one wishes to construct some examples of an equation associated to a
given affine Weyl group one has to specify a nonclosed periodically repeated
pattern in the appropriate space, and moreover since any such pattern would
lead to a discrete Painlev´e equation the potential number of discrete Painlev´e
equations in infinite.
On the other hand, since the continuous Painlev´e equations appeared as
similarity reductions of soliton equations, it is natural to think about the
same thing for similarity reduction of lattice equations. In [44] we discussed
various travelling wave reductions of the deautonomised classical discrete
soliton equations (KdV, mKdV, SG and Burgers). It was shown that indeed various discrete Painlev´e equations are obtained. Also in [45] the same
approach has been done on the Lax pair of nonautonomous mKdV and a
lot of Painlev´e equations appeared. In this direction the travelling wave reduction applied to the famous Adler-Bobenko-Suris cube-consistent lattice
equations [46] is quite tempting and a lot of results appeared. The case of
Q4 lattice equation is rather special. It is in fact the master equation for the
ABS-classification and moreover is an integrable discretisation of the famous
Krichever-Novikov equation [47]. All the other equations in ABS class appear as a result of a degeneration cascade. Sakai showed that corresponding
to Kodaira’s elliptic singular fibers the discrete equations can be classified
in elliptic, q-discrete and difference type equations. In particular the elliptic
(1)
equations are related to automorphisms of surfaces of A0 - type (I0 in Kodaira classification). We are going to show that the travelling wave reduction
of Q4 ABS- lattice equation can be lifted to an automorphism of a rational
(1)
elliptic surface having A1 type fibers. Accordingly the corresponding nonau52
tonomous equations can be only multiplicative or additive but not elliptic as
it was suggested by deautonomisation using singularity confinement.
In order to get a clear description of quadrilateral lattice equations Adler,
Bobenko and Suris proposed a classification based on a special symmetry
namely consistency around the cube[46]. This allows to construct immediately the discrete zero curvature representation thus proving the integrability.
Up to homographic and linear transformations a part((Q-list) of the quadrilateral lattice equations were classified as follows: (for simplicity we use the
notations x = xn,m , x¯ = xn+1,m , x˜ = xn,m+1 , etc.)
Q4:
sn(α; k)(xx˜¯ + x¯x˜) − sn(β; k)(x¯
x + x˜x˜¯)−
− sn(α−β; k)(x˜
x + x¯x˜¯)+sn(α; k) sn(β; k) sn(α−β; k)(1+k 2 x¯
xx˜x˜¯) = 0 (7.1)
In the case k → 0 then the elliptic sin goes to ordinary sin and Q4 →
Q3(below)
Q3:
sin α(xx˜¯ + x¯x˜) − sin β(x¯
x + x˜x˜¯)−
Q2:
− sin(α − β)(x˜
x + x¯x˜¯) + sin α sin β sin(α − β) = 0
a(x − x˜)(¯
x − x˜¯) + b(x − x¯)(˜
x − x˜¯)+
+c(x + x˜ + x¯ + x˜¯) + d = 0
(7.2)
(7.3)
where c, d are expressed in terms of a and b
Q1:
α(x − x˜)(¯
x − x˜¯) + β(x − x¯)(˜
x − x˜¯) + δ = 0
(7.4)
These equations form a degeneration cascade; If, sinα = a, sin β = b, x →
1 + ǫx, sin(α − β) = −(a + b) + ǫc, sin(α − β) sin α sin β = −2ǫc + ǫ2 d, then
Q3 → Q2
In order to obtain a mapping we make the so called (p,q)-reduction≡
travelling wave reduction, namely:
xn,m = xpn+qm = xν
The simplest reduction appears for the travelling wave with speed 1: xn,m+1 =
xn+1,m . In this case the Q4 mapping becomes:
53
(sn(α; k) − sn(β; k))(x¯
x + xx)−
− sn(α − β; k)(¯
xx + x2 ) + sn(α; k) sn(β; k) sn(α − β; k)(1 + k 2 x2 x¯x) = 0
It can be written as φ : P1 × P1 → P1 × P1 in the form:
x¯ = y
y¯ =
and also the inverse:
By 2 − Gxy − A
Ak 2 xy 2 − Bx + Gy
y=x
x=
Bx2 − Gxy − A
Ak 2 yx2 − By + Gx
where A, B, G are expressed in terms of elliptic Jacobi sines. The blow up
points can be computed from the expressions but unfortunately are quite
complicated. In order to get through this we change the parametrisation.
Namely we introduce variables γ, z by α = γ + z, β = γ − z. Using addition
formulas for elliptic functions we obtain:
A = (cn2 (z; k) − cn2 (γ; k)) cn(z; k) dn(z; k)
B = cn(z; k) dn(z; k)(1 − k 2 sn2 (γ; k) sn2 (z; k))
G = cn(γ; k) dn(γ; k)(1 − k 4 sn4 (z; k))
7.1
Resolution of singularities and symmetry group
In this parametrisation we define the space of initial conditions as a rational
surface X obtained after blow ups of the following 8 points (Ei , i = 1...4 are
indeterminate points for φ and Ej , j = 5, ..., 8 for φ−1 :
E1 : (x, y) =
E3 : (x, y) =
cn(γ) cn(z)
,
dn(γ) dn(z)
dn(γ) dn(z)
,
k cn(γ) k cn(z)
, E2 : (x, y) =
, E4 : (x, y) =
54
cn(γ)
cn(z)
−
,−
dn(γ) dn(z)
dn(γ)
dn(z)
−
,−
k cn(γ) k cn(z)
cn(γ)
cn(z)
,−
, E6 : (x, y) = −
E5 : (x, y) =
dn(z) dn(γ)
dn(z) dn(γ)
dn(z)
dn(γ)
E7 : (x, y) =
, E8 : (x, y) = −
,
,−
k cn(z) k cn(γ)
k cn(z) k cn(γ)
cn(z) cn(γ)
,
dn(z) dn(γ)
After the blowing up points Ei the mapping is lifted to φ : X → P1 × P1
which is free of any singularities. Also one can check by direct (and long)
calculation that φ : X → X and its inverse are free of any singularities.
Accordingly the mapping is an automorphism of a ratuional surface.
Now we are going to show the action on the Picard group. First of all it
is easily seen that the image of the Ej , j = 5, ..., 8 are Ei , i = 1...4 namely
(for convenience we note φ(Ei ) as E¯i ):
E¯5 = E1 , E¯6 = E2 , E¯7 = E3 , E¯8 = E4
For the the image of the total transform of the line x = 0.
(¯
x, y¯)|x=0,y = (y,
By 2 − A
)
Gy
which is the curve Bx2 −Gxy−A passing through E5 , E6 , E7 , E8 . Accordingly
H¯x = 2Hx + Hy − E5 − E6 − E7 − E8
H¯y = Hx
In the same way we get the following:
Hy − E 1 → E 5 → E 1 → Hx − E 5
Hy − E 2 → E 6 → E 2 → Hx − E 6
Hy − E 3 → E 7 → E 3 → Hx − E 7
Hy − E 4 → E 8 → E 4 → Hx − E 8
This is exactly the singularity confinement pattern. It shows a strictly confining shape (guaranteed by the integrability of the mapping)
Let us denote anPelement of the Picard lattice < Hx , Hy , E1 ...E8 >Z by
A = h0 Hx + h1 Hy + 8i=1 ei Ei (hi , ej ∈ Z), then the induced bundle mapping
55
is acting on it as
φ∗ (h0 , h1 , e1 , ..., e8 )








=(h0 , h1 , e1 , ..., e8 ) 







2
1
1
1
1
1
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0

0 −1 −1 −1 −1
0 0
0
0
0 

0 −1 0
0
0 

0 0 −1 0
0 

0 0
0 −1 0 

0 0
0
0 −1 

0 0
0
0
0 

0 0
0
0
0 

0 0
0
0
0 
1 0
0
0
0
From the eigenspace of value 1 eigenvector we get the invariant of the mapping. It turns out that the anticanonical divisor is preserved
¯ X = −KX = 2Hx + 2Hy −
−K
8
X
Ei
i=1
. The proper transform of the anticanonical divizor gives the following pencil
of elliptic curves (λ parametrizes the elliptic fibration )
f0 (x, y) + λg0 (x, y) ≡ (1 + k 2 x2 y 2 )(AB − λAG) − (x2 + y 2 )(B 2 + λGB)+
+2xy(BG − λk 2 A2 + λB 2 ) = 0
From the fact that:
¯ = −λ
λ
the corresponding invariant is given by λ2 = (f0 (x, y)/g0 (x, y))2 . Also because degf0 =degg0 = (2, 2) we can conclude that the mapping belongs to the
case (ii-1) in our classification and exchange fibers in a Halphen surface of
index one.
In order to see what is the type of the surface X we take the following
curves:
cn(z)
cn(γ) cn(z)
cn(γ) cn(z)
cn(γ)
−
k−1 +
−
=0
+(x−y)
D1 = −xyk
dn(γ) dn(z)
dn(γ) dn(z)
dn(γ) dn(z)
56
D2 = −xyk
cn(z)
cn(γ) cn(z)
cn(γ) cn(z)
cn(γ)
−
k−1 +
−
=0
−(x−y)
dn(γ) dn(z)
dn(γ) dn(z)
dn(γ) dn(z)
Their proper transforms are given by:
D1 = H x + H y − E 1 − E 3 − E 6 − E 8
D2 = H x + H y − E 2 − E 4 − E 5 − E 7
and
−KX = D1 + D2
Also the action of the mapping permutes the curves D¯1 = D2 , D¯2 = D1
and the intersection form is D1 · D2 = 2. Accordingly the surface X is an
(1)
A1 -type.
The affine Weyl group symmetries are related to the orthogonal complement of Dred = {D1 , D2 }. In order to see this, we note that rank Pic(X) =
rank hHx , Hy , E1 , ...E8 iZ = 10. The orthogonal complement of Dred :
hDi⊥ = {α ∈ Pic(X)|α · Di = 0, i = 0, 3}
has 8-generators :
hDi⊥ = hα1 , α1 , ..., α8 iZ , with
α1 = E 1 − E 3 , α2 = E 3 − E 6 , α3 = E 6 − E 8
α4 = E 2 − E 4 , α5 = E 4 − E 5 , α6 = E 5 − E 7
α7 = Hx − Hy , α8 = Hy − E 2 − E 8
Now the family hα1 , α1 , ..., α8 i together with the intersection form of divisors
can be seen as a root lattice associated to a Weyl group. Computing the
Cartan matrix cij = 2(αi · αj )/(αi · αi ) one gets the structure of affine Weyl
(1)
group E7 . As usual the mapping can be written in terms of elementary re˜ (E7(1) . This fact suggests
flections associated to the extended Weyl group W
that the mapping cannot be the autonomous of an elliptic Painlev´e equation.
However because the decomposition of the anticanonical divisor in effective
divisors is not necessarily unique it may be possible to find another decom(1)
position which provides a fully elliptic surface A0 . To our knowledge this
has not been done so far and we strongly believe that our decomposition is
the only one.
57
8
Minimization of elliptic surfaces from birational dynamics
As we said [63] Diller and Favre showed that for any birational automorphism
ϕ on a projective smooth rational surface S, we can construct a rational
surface S˜ by successive blow-ups from S such that (i) σ ◦ ϕ˜ = ϕ ◦ σ, where σ
denotes the successive blow-downs σ : S˜ → S, (ii) ϕ˜ = ϕ on S in generic, and
(iii) ϕ˜ : S˜ → S˜ is analytically stable. In general, ϕ˜ is said to be lifted from ϕ
if the condition (i) and (ii) are satisfied and a birational automorphism ϕ on
S is said to be analytically stable if the condition ((ϕ)∗ )n = ((ϕ)n )∗ holds on
the Picard group on S. The notion of analytical stability is closely related to
the singularity confinement. Indeed, this notion is equivalent to the condition
that there is no curve C on S and a positive integer k such that ϕ(C) is a
point on S and ϕk (C) is an indeterminate point of ϕ, i.e. analytical stability
demands that singularities are not recovered by the dynamical system. In
other words, if a mapping ϕ satisfies singularity confinement criterion, i.e.
for any curve such that ϕ(C) is a point, ϕk+1 (C) recovers to a curve again
for some positive integer k, then if we blow up the phase space at ϕi (C) for
1 ≤ i ≤ k, then the singularity would be relaxed and resolved by successive
applications of this procedure. Finally we would obtain a surface where the
lifted birational automorphism ϕ˜ is analytically stable.
From Diller and Favre’s work, such birational automorphisms f are classified as follows: Let f be a bimeromorphic automophism of a K¨ahler surface
with the maximum eigenvalue of f ∗ is one. Up to bimeromorphic conjugacy,
exactly one of the following holds.
• The sequence ||(f n )∗ || is bounded, and f n is an automorphism isotopic
to the identity for some n, where ||·|| denotes the Euclidean norm w.r.t.
some basis of the Picard group.
• The sequence ||(f n )∗ || grows linearly, and f preserves a rational fibration. In this case, f can not be conjugated to an automorphism. (We
say f is linearizable or linearizable in cascade in this case [56, 72]).
• The sequence ||(f n )∗ || grows quadratically, and f is an automorphism
preserving an elliptic fibration.
These three conditions are essential in analysing any birational dynamical system on P2 . They represent the rigurous formulation of the complexity
58
growth (algebraic entropy) criterion) exposed at the beginnig of the thesis.
In addition these conditions show that a secific growth provides also the
method of integration. Namely if the growth is linear the system is linearisable preserving a rational fibration. But because it cannot be conjugated to
an automorphism the number of blowing ups can be infinite (we shall return
to the problem of linearisable systems). If the growth is quadratic then the
mapping can be integrated in terms of elliptic functions since preserves an
elliptic fibration (and the number of blow ups is finite). For exponential
growth we have the chaotic case. So the whole analysis of complexity growth
can be done also using the bundle mapping on the Picard group.
Though in this paper we consider mainly autonomous case, the procedure constructing analytically stable mapping can be applied also for nonautonomous case such as linearizable mappings or discrete Painlev´e equations. In this case, we start from a sequence of birational mappings ϕi :
S → S and blow up successively at confined singular points whose positions depend on i. Then, ϕi would be lifted to a sequence of birational
mappings ϕ˜i : S˜i → S˜i+1 such that (i) σi+1 ◦ ϕ˜i = ϕi ◦ σi , (ii) ϕ˜i = ϕi
on S in generic, and (iii) {ϕ˜ : S˜i → S˜i+1 }i∈Z is analytically stable, i.e.
ϕ∗i ◦ · · · ◦ ϕ∗i+n = (ϕi+n ◦ · · · ◦ ϕi )∗ holds on the Picard group on S˜i+n+1 for
any i and non-negative integer n (see [58, 59, 72] about computation and the
relation to the degree growth).
In the study of integrable systems, we often want to find conserved quantities or linearize a given integrable mappings, but the above construction of
analytically stable mapping does not guarantees that
(i) ϕ˜ is an automorphism.
˜
(ii) S˜ is relatively minimal, i.e. there does not exists a blow-down of S,
π : S˜ → S˜′ such that ϕ˜′ is still analytically stable on S˜′ .
In other words, the following possibilities remain.
(ia) A singularity sequence consists of infinite sequences of points to both
sides and a finite sequence of curves:
· · · → point → point → curves → · · · → curves → point → point → · · · ,
where the image of a curve C, parametrized as (f (t), g(t)) on some coordinates, under ϕn is defined as the Zariski closure of limε→0 ϕn (f (t)+c1 ǫ, g(t)+
c1 ǫ) with generic t, c1 and c2 .
(ib) A singularity sequence consists of an infinite sequence of points and that
59
of curves to each side:
· · · → point → point → · · · · · · → curves → curves → · · · .
(ii)’ A finite set of exceptional curves are permuted.
Proposition 1.7 and Lemma 4.2 of [63] (cf. [51]) says that curves in
(ia) can be blown down, and that (ib) occurs only if f is not conjugate to an
automorphism, i.e. if f is linearizable or has a positive entropy. And if a curve
in Case (ii)’ or Case (ib) is exceptional of the first kind, we can blow down
them. Hence theoretically, we can obtain relatively minimal analytically
stable surfaces and compute the action on the Picard group. However, for
investigating properties of the mapping f such as conserved quantities, we
need to know coordinate change explicitly.
Our aim in this part is to develop a method to control blowing down structures on the level of coordinates. We apply our method to various examples,
including the newly studied discretization of reduced Nahm equations [55].
In general, finding elliptic fibration for an elliptic surface is not easy if the
surface is not minimal, and we use information of singularity patterns of the
dynamical systems for finding unnecessary (−1) curves. Accordingly, this
method of minimization shows how an order-two mapping with complicated
singularity structure can be brought to a simpler form which enables computations of conserved quantities.
In the next section, we recall some basic notions and blowing down structures. Then, we investigate discrete versions of reduced Nahm equations,
which preserve a rational elliptic fibration. We will show that the associated surfaces are not minimal and by minimization one can transform the
mappings to simpler ones. Further on we investigate linearizable dynamical
systems, including non-autonomous case.
60
8.1
Blowing down structure
Notations: cf. [52, 53]
S:
D:
D · D′ :
O(D) :
Pic(S) =
≃
a smooth rational surface
the linear equivalent class of a divisor D
the intersection number of divisors D and D′
the invertible sheaf corresponding to D
the group of isomorphism classes of invertible sheaves on S
the group of linear equivalent classes of divisors on S
E : the total transform of divisor class of a line on P2
Hx , Hy : the total transform of divisor class of a line x = constant
(or y = constant) on P1 × P1
Ei : the total transform of the exceptional divisor class of the i-th blow-up
|D| ≃(H 0 (S, O(D)) − {0})/ C× : the linear system of D
KS : the canonical divisor of a surface S
g(C) : the genus of an irreducible curve C, given by the genus formula
g(C) = 1 + 21 (C 2 + C · KS ) if C is smooth.
Let S = Sm be a surface obtained by successive m times blowing up from
P (or any rational surface) at indeterminate or extremal point of ϕ, i.e. the
Jacobian ∂(¯
x, y¯)/∂(x, y) in some local coordinates is zero, such that ϕ˜ on §
is analytically stable. Let Fm be a curve on S with self-intersection −1 and
Fm be the corresponding divisor class. Our strategy to write the blow-down
Sm along Fm by coordinates is as follows.
Take a divisor class F such that there exists a blowing down structure (this
terminology is due to [50]): S = Sm → Sm−1 → Sm−2 → · · · → S1 → P2 ,
where Sm → Sm−1 is a blow-down along Fm and each Si → Si−1 is a blowdown along an irreducible curve, such that the divisor class of lines in P2 is
F. Let |F| = α0 f0 + α1 f1 + α2 f2 = 0. Then (f0 : f1 : f2 ) gives P2 coordinates.
In order to find such F we note the following facts.
It is necessary for the existence of such a blow-down structure that there
exists a set of divisor classes F1 , . . . , Fm such that
2
F2 = 1 Fi2 = −1,
Fi · Fj = 0,
61
F · Fi = 0
for (1 ≤ i, j ≤ m), and further that (i) the genus of divisor F is zero; (ii)
the linear system of F does not have a fixed part in the sense of Zariski
decomposition and its dimension is two.
If the linear system of F does not have fixed part, then by Bertini theorem,
its generic divisor is smooth and irreducible (this follows from the fact that
two divisors defines a pencil by blowing up at the unique intersection and P.
137 of [53]), and its genus is given by the formula
1
g = 1 + (F 2 + F · KS ).
2
From this fact and Condition (ii), 1 + 12 (F 2 + F · KS ) should be zero.
Example 8.1. If degree of F is less than 6, then F is given by one of the
following forms.
E
2E − Ei1 − Ei2 − Ei3
3E − 2Ei1 − Ei2 − Ei3 − Ei4 − Ei5
4E − 2Ei1 − 2Ei2 − 2Ei3 − Ei4 − Ei5 − Ei6
4E − 3Ei1 − Ei2 − Ei3 − Ei4 − Ei5 − Ei6 − Ei7
5E − 2Ei1 − 2Ei2 − 2Ei3 − 2Ei4 − 2Ei5 − 2Ei6
5E − 3Ei1 − 2Ei2 − 2Ei3 − 2Ei4 − Ei5 − Ei6 − Ei7
5E − 4Ei1 − Ei2 − Ei3 − Ei4 − Ei5 − Ei6 − Ei7 − Ei8 − Ei9 ,
(8.1)
where ij ’s are all distinct with each other. All the above F admit blow-down
structure if the positions of blow-up points are generic. For example, for
F = 2E − Ei1 − Ei2 − Ei3 , Fi ’s are given by
E − Ei − Ej ({i, j|i 6= j} ⊂ {i1 , i2 , i3 }), Ej (j 6= i1 , i2 , i3 )
and for F = 3E − 2Ei1 − Ei2 − Ei3 − Ei4 − Ei5 , Fi ’s are given by
E−Ei1 −Ej (j ∈ {i2 , . . . , i5 }), 2E−Ei1 −Ei2 −Ei3 −Ei4 −Ei5 , Ej (j 6= i1 , . . . , i5 ).
If we want to blow down to P1 × P1 instead of P2 , our strategy becomes
as follows.
Let Fm−1 be a curve on S with self-intersection −1 and Fm−1 be the
corresponding divisor class. Take a divisor class Hu and Hv such that there
exists a blow-down structure: S = Sm−1 → Sm−2 →→ · · · → S1 → P1 × P1 ,
62
where Sm−1 → Sm−2 is a blow-down along Fm−1 and each Si → Si−1 is a
blow-down along an irreducible curve, such that the divisor class of lines
u = const and v = const are Hu and Hv . Let |Hu | = α0 f0 + α1 f1 = 0
and |Hv | = β0 g0 + β1 g1 = 0. Then (u, v) = (f0 /f1 , g0 /g1 ) gives P1 × P1
coordinates.
In this case, it is necessary that there exits a set of divisor classes F1 , . . . , Fm−1
such that
Hu2 = Hv2 = 0, Hu · Hv = 1, Fi2 = −1,
Fi · Fj = 0, Hu · Fi = Hv · Fi = 0
for (1 ≤ i 6= j ≤ m − 1), and further that (i) each genus of divisor Hu or Hv
is zero; (ii) each linear system of Hu or Hv does not have a fixed part and
its dimension is one. Consequently, 1 + 12 (F 2 + F · KS ) should be zero again.
Example 8.2. If S is obtained by successive blow-ups from P2 , and the sum
of degree of Hu or Hv is less than 6, then each Hu or Hv is given by F − Ek ,
where F is in the list (8.1).
If S is obtained by successive blow-ups from P1 × P1 , each Hu or Hv is
given by
Hx
H x + H y − Ei1 − Ei2
2Hx + Hy − Ei1 − Ei2 − Ei3 − Ei4
2Hx + 2Hy − 2Ei1 − Ei2 − Ei3 − Ei4 − Ei5
3Hx + Hy − Ei1 − Ei2 − Ei3 − Ei4 − Ei5 − Ei6
3Hx + 2Hy − 2Ei1 − 2Ei2 − Ei3 − Ei4 − Ei5 − Ei6
4Hx + Hy − Ei1 − Ei2 − Ei3 − Ei4 − Ei5 − Ei6 − Ei7 − Ei8
(8.2)
and those with exchange of Hu and Hv . Not all, but many pairs of these
divisor classes admit a blow-down structure for generic blow-up points. For
example, for Hu = Hx and Hu = Hx + Hy − Ei1 − Ei2 , Fi ’s are given by
Hx − Ei1 , Hx − Ei2 Ej (j 6= i1 , i2 )
and for Hu = Hx + Hy − Ei1 − Ei2 and Hu = Hx + Hy − Ei1 − Ei3 , Fi ’s are
given by
Hx − Ei1 , Hy − Ei1 , Hx + Hx − Ei1 − Ei2 − Ei3 , Ej (j 6= i1 , i2 , i3 ).
63
Remark 8.3. There is another way to obtain relatively minimal surface for
elliptic surface case, though it needs heavy computation. Let S be a rational
elliptic surface (not necessarily minimal) where the mapping ϕ is lifted to an
automorphism. Compute a R-divisor θ by
ϕ˜n∗ (E)
,
n→∞ ||ϕ
˜n∗ (E)||
θ := lim
where || · || denotes the Euclidian norm of a divisor w.r.t. a fixed basis, and
let k > 0 be a minimum number such that kθ ∈ Pic(S). Then, the linear
system |mkθ| gives an elliptic fibration for some integer m ≥ 1 (m is not
always one, (cf. Step 1 of Appendix of [63] and [2]). Let C be a curve in the
linear system |kθ| (such C exists [62]). By applying van Hoeji’s algorithm
[69] (cf. [60]), we obtain a birational transformation S → S ′ , (x, y) 7→ (u, v)
such that C is transformed into Weierstrass normal form v 2 = u3 − g2 u − g3 .
Since the degree of this curve is three, S ′ is obtained by 9 blow-ups from
P2 . This implies S ′ is a minimal elliptic surface (the fibration is given by the
linear system | − mKS ′ |).
Remark 8.4. If ϕ is an automorphism of a non-minimal rational elliptic
surface, the invariant does not corresponds to the anti-canonical divisor, because the self-intersection of the anti-canonical divisor is negative in this case,
while θ2 of the above remark should be zero.
8.2
A simple example which needs blowing down
Let us show first a simple example which needs change of blow-down structure
to obtain relatively minimal surface. This example is due to Diller and Favre’s
paper [63]: (for simplicity we note xn = x, x¯ = xn+1 , x = xn−1 and so forth)

1


 x¯ = y +
2
.
(8.3)
x(2y
− 1)


 y¯ =
2y + 2
This system can be lifted to an automorphism on a surface S by blowing up
P1 × P1 at the singularity points of the dynamical systems:
E1 : (x, y) = (1, 0), E2 (1/2, −1/2), E3 (0, −1), E4 (−1/2, ∞),
E5 (∞, −1/2), E6 (0, ∞), E7 (∞, 0), E8 (1/2, ∞), E9 (∞, 1/2).
64
Immediately one can see the action on the Picard group from the following
singularity patterns:
H y − E3 → E4 → E5 → E6 → E7 → E8 → E9 → H x − E1
H y − E9 → E1 → E2 → E3 → H x − E4
and also the invariant divisor classes Hx + Hy − E1 − E2 − E3 and
Hx + Hy − E4 − E5 − E6 − E7 − E8 − E9 . The presence of invariant divisor
calsses imposes making blow-down along the curve which corresponds to
the divisor class Hx + Hy − E1 − E2 − E3 (it is the only one which has selfintersection -1, the other has self-intersection -3). Hence we take the basis of
blow-down structure as
H u = H x + H y − E2 − E 3 , H v = H x + H y − E1 − E2 ,
H x + H y − E 1 − E2 − E 3 , F 1 = H x − E 2 , F 2 = H y − E 2 ,
Fi = Ei+1 (i = 3, 4, 5, 6, 7, 8),
where the linear systems of Hu and Hv are given by
| Hu | : u0 (x − y − 1) + u1 (2xy + x) = 0,
| Hv | : v0 (x − y − 1) + v1 (2xy − y) = 0.
Using these, we take the following change of variables:
u=
2xy + x
,
x−y−1
v=
2xy − y
,
x−y−1
then our dynamical system (3) and (4) becomes

2uv − u − v − 1

 u¯ =
u − 3v + 1
.
−2uv

 v¯ =
u+v+1
(8.4)
This system has the following blow-up points:
F1 : (u, v) = (−1, 0), F2 (0, −1), F3 (1, 2), F4 : (u, (v + 1)/u) = (0, 1).
F5 (0, 1), F6 (1, 0), F7 : ((u + 1)/v, v) = (1, 0), F8 (2, 1).
and the linear system of the anti-canonical divisor gives the invariant
K=
uv(2uv − u − v − 1)
x(2x − 1)y(2y − 1)(2xy − x + y + 1)
=
2
(u − v) − 1
(x − y − 1)2
65
and the invariant two form
ω=
8.3
dx ∧ dy
du ∧ dv
=
.
2
(u − v) − 1
1−x+y
Discrete Nahm equations with tetrahedral symmetry
In [55], Petrera, Pfadler and Suris proposed the following discretization of
the reduced Nahm equations with tetrahedral symmetry
x¯ − x = ǫ(x¯
x − y y¯)
.
(8.5)
y¯ − y = −ǫ(x¯
y + y¯
x)
Here ǫ is related to the step of discretization. The integrability can be proved
by the existence of the following conserved quantity and invariant two-form
K=
y(3x2 − y 2 )
,
−1 + ǫ2 (x2 + y 2 )
ω=
dx ∧ dy
.
y(3x2 − y 2 )
(8.6)
In this case one can easily transform the system into a QRT one by the
following variable transformation
u=
1 + ǫx
1 − ǫx
,v =
.
y
y
(8.7)
Immediately we get u¯ = v. From the equation (8.5) we get a QRT mapping
3¯
uu − u(¯
u + u) − u2 + 4ǫ2 = 0
with the invariant:
K=
−3(u − v)2 + 4ǫ2
,
2ǫ2 (u + v)(uv − ǫ2 )
ω=
du ∧ dv
,
3(u − v)2 − 4ǫ2
which are precisely (8.6) in the variables x and y.
Now we are going to study the singularity structure and its space of initial
conditions and recover the invariants. The fact that the conserved quantity
is expressed by a ratio of a cubic polynomial implies that we have better to
start with P2 than P1 × P1 .
On P2 : (X : Y : Z) = (x : y : 1), we blow up the following points
66
√
√
√
E1 (−1 : − 3 : 2ǫ), E2 (1 : 3 : 2ǫ), E3 (−1 : 3 : 2ǫ),
√
E4 (1 : − 3 : 2ǫ), E5 (1 : 0 : ǫ), E6 (−1 : 0 : ǫ),
E7 (1 : 0 : 0), E8 (1 : 1 : 0), E9 (1 : −1 : 0).
In order to blow down to P1 × P1 , we take the basis of blow-down structure
Hx , Hy , F1 , . . . , F8 as
Hx = E − E5 , Hy = E − E6 , Fi = Ei (i = 1, 2, 3, 4),
F5 = E7 , F6 = E8 , F7 = E8 , F8 = E − E5 − E7 .
The curves corresponding to the divisor classes Hx and Hy are:
α0 (ǫX − Z) + α1 Y = 0,
β0 (ǫX − Z) + β1 Y = 0.
They give immediately the change of variable
u=
ǫx − 1
,
y
v=
ǫx + 1
,
y
which is essentially (8.6) up to rescaling factors.
8.4
Discrete Nahm equations with octahedral symmetry:
The second Nahm equation is the one corresponding to octahedral symmetry.
The system has the following form
x¯ − x = ǫ(2x¯
x − 12y y¯)
,
(8.8)
y¯ − y = −ǫ(3x¯
y + 3y¯
x + 4y y¯)
which is again integrable by the invariants:
y(2x + 3y)(x − y)2
1 − 10ǫ2 (x2 + 4y 2 ) + ǫ4 (9x4 + 272x3 y − 352xy 3 + 696y 4 )
dx ∧ dy
ω=
.
y(x − y)(2x + 3y)
K=
67
(8.9)
Inspired by the transformation (8.7) we can simplify the system by the following transformations:
1
x = (χ − 2y),
3
1
x¯ = (χ¯ − 2¯
y)
3
and u = (1 − ǫχ)/y, v = (1 + ǫχ)/y. Finally we get a simpler equation but
non-QRT type:
u + u) + 20ǫ(¯
u − u) − 4u2 + 400ǫ2 = 0,
8¯
uu − 2u(¯
which can be written as a system on P1 × P1

 u¯ = v
(u + 2v − 20ǫ)(v + 10ǫ) .
 v¯ =
4u − v + 10ǫ
(8.10)
The space of initial conditions is given by the P1 × P1 blown up at the
following nine points:
E1 : (u, v) = (−10ǫ, 0), E2 (0, 10ǫ), E3 (10ǫ, 5ǫ),
E4 (5ǫ, 0), E5 (0, −5ǫ), E6 (−5ǫ, −10ǫ)
E7 (∞, ∞), E8 : (1/u, u/v) = (0, −1/2), E9 : (1/u, u/v) = (0, −2).
The action on the Picard group is the following:
H¯u = 2 Hu + Hv − E1 − E3 − E7 − E8 , H¯v = Hu
E¯1 = E2 , E¯2 = Hu − E3 , E¯3 = E4 , E¯4 = E5 , E¯5 = E6 ,
E¯6 = Hu − E1 , E¯7 = Hu − E8 , E¯8 = E9 , E¯9 = Hu − E7 .
From this action one can see immediately that we have three invariant divisor
classes:
α 0 = H u + H v − E1 − E 2 − E7 , α 1 = H u + H v − E 1 − E2 − E8 − E 9 ,
α 2 = E7 − E8 − E 9 , α 3 = H u + H v − E3 − E 4 − E5 − E6 − E 7 .
The curve corresponding to α0 is a (-1) curve which must be blown down.
Let Ha = Hu + Hv − E2 − E7 and Hb = Hu + Hv − E1 − E7 , then their linear
systems are given by
a1 u + a2 (v − 10ǫ) = 0,
68
b1 (u + 10ǫ) + b2 v = 0
and the basis of blow-down structure is given by
H a , H b , α 0 , F 1 = H u − E7 , F 2 = H v − E7 ,
F3 = E3 , F4 = E4 , F5 = E5 , F6 = E6 , F7 = E8 , F8 = E9 .
So if we set:
u + 10ǫ
v − 10ǫ
b=
,
u
v
our dynamical system becomes

3ab − 2a + 2

 a
¯ =
a−4
.
 ¯b = 4 − a

2a + 1
a=
(8.11)
This system has the following space of initial conditions which define a
minimal rational elliptic surface:
F1
F3
F5
F7
: (a, b) = (0, ∞), F2 : (a, b) = (∞, 0),
: (a, b) = (−1/2, 4), F4 : (a, b) = (−2, ∞)
: (a, b) = (∞, −2), F6 : (a, b) = (4, −1/2),
: (a, b) = (−2, −1/2), F8 : (a, b) = (−1/2, −2).
The invariants can be computed from the anti-canonical divisor as
K=
da ∧ db
(ab − 1)(ab + 2a + 2b − 5)
, ω=
4ab + 2a + 2b + 1
(ab − 1)(ab + 2a + 2b − 5)
which are equivalent to the invariants (8.9).
8.5
Discrete Nahm equations with icosahedral symmetry
The last example of discrete reduced Nahm equations refers to icosahedral
symmetry. It is given by
x¯ − x = ǫ(2x¯
x − y y¯)
(8.12)
y¯ − y = −ǫ(5x¯
y + 5y¯
x − y y¯)
69
and is integrable as well. However the invariants here are more complicated.
They are reported also by [55] as1
K=
y(3x − y)2 (4x + y)3
,
1 + ǫ2 c 2 + ǫ4 c 4 + ǫ6 c 6
ω=
dx ∧ dy
y(3x − y)(4x + y)
(8.13)
where
c2 = −7(5x2 + y 2 )
c4 = 7(37x4 + 22x2 y 2 − 2xy 3 + 2y 4 )
c6 = −225x6 + 3840x5 y + 80xy 5 − 514x3 y 3 − 19x4 y 2 − 206x2 y 4 .
Again we can make first the following change of variable
1
y
x = (X + ),
5
2
1 ¯ y¯
x¯ = (X
+ ),
5
2
then we divide by y y¯ both equations and call again a = X/y, b = 1/y, u =
b − ǫa, v = b + ǫa and finally we get a simpler equation but non-QRT type:
u + u) −
6¯
uu − u(¯
7ǫ
(¯
u − u) − 4u2 + 49ǫ2 = 0.
2
We can apply our procedure to this last non-QRT mapping. However,
here we demonstrate that our procedure works well even for the original
mapping.
The space of initial condition is given by the P1 × P1 blown up at the
following 12 points:
E1 : (x, y) = (∞, ∞), E2 (−1/7ǫ, −3/7ǫ), E3 (−1/7ǫ, 4/7ǫ),
E4 (1/7ǫ, 3/7ǫ), E5 (1/7ǫ, −4/7ǫ) E6 (1/5ǫ, 0),
E7 (1/3ǫ, 0), E8 (1/ǫ, 0), E9 (−1/ǫ, 0),
E10 (−1/3ǫ, 0), E11 (−1/5ǫ, 0), E12 : (1/x, x/y) = (0, 1/3).
On this surface the dynamical system is neither an automorphism nor
analytically stable due to the following topological singularity patterns:
Hy − E1 (y = ∞) → point → · · · (4 points) · · · → point → Hy − E1
· · · → point → point → Hx − E1 (x = ∞) → point → point → · · · ,
1
a sign in c2 was corrected by information from the authors of that paper
70
where the image of a curve under ϕn is defined as (ia) in Section 1. Moreover,
the curve 4x + y = 0 : Hx + Hy − E1 − E3 − E5 is invariant. We blow down
along these three curves with the blow-down structure
H u = H x + H y − E 1 − E3 , H v = H x + H y − E1 − E5 ,
H x − E 1 , H y − E 1 , H x + H y − E 1 − E 3 − E5 ,
F1 = E12 , F2 = E2 , F3 = E4 , F4 = E6 ,
F5 = E7 , F6 = E8 , F7 = E9 , F8 = E10 , F9 = E11 ,
where the linear systems of Hv and Hv are given by
| Hu | :u0 (1 + 7ǫx) + u1 (4x + y)
| Hv | :v0 (1 − 7ǫx) + v1 (4x + y).
If we take the new variables u and v as
v=
2(1 − 7ǫx)
2(1 + 7ǫx)
, v=
,
ǫ(4x + y)
ǫ(4x + y)
then we have
F1 : (u, v) = (2, −2), F2 : (0, −4), F3 : (4, 0), F4 : (6, −1), F5 : (5, −2),
F6 : (4, −3), F7 : (3, −4), F8 : (2, −5), F9 : (1, −6).
The dynamical system becomes an automorphism having the following topological singularity patterns
Hv − F 9 → F 2 → F 1 → F 3 → Hu − F 4
Hv − F 3 → F 4 → F 5 → F 6 → F 7 → F 8 → F 9 → Hu − F 2
and Hu → Hu + Hv − F2 − F4 . Hence we find the invariant (−1) curve
Hu + Hv − F1 − F2 − F3 , which should be blown down. Again we take the
blow-down structure as
Hs = Hu + Hv − F 1 − F 2 , Ht = Hu + Hv − F 1 − F 3 ,
Hu + Hv − F1 − F2 − F3 , F′1 = Ha − F1 , F′2 = Hb − F1
F′3 = F4 , F′4 = F5 , F′5 = F6 , F′6 = F7 ,
F′7 = F8 , F′8 = F9 ,
71
where the linear systems of Hs and Ht are given by
| Hs | :s0 u(v + 2) + s1 (u − v − 4)
| Ht | :t0 v(u − 2) + t1 (u − v − 4)
and hence we take the new variables s and t as
s=−
3v(u − 2)
3u(v + 2)
, t=−
.
2(u − v − 4)
2(u − v − 4)
Then we have
F′1 : (s, t) = (3, 0), F′2 (0, 3), F′3 (−3, 2), F′4 : (
F′5 (2, 3), F′6 (3, 2), F′7 : (u − 3,
and
s
, d − 3) = (5, 0),
t−3
t
) = (0, 5), F′8 (2, −3)
s−3

2st − 3s − 3t + 9

 s¯ =
s+t−3
.
2(s
−
3)(t + 3)

 t¯ =
3s − t − 9
The invariants can be computed by using the the anticanonical divisor as
K′ =
(s − t)2 + 4(s + t) − 21
−56ǫ6 y(−3x + y)2 (4x + y)3
=
(s − 2)(t − 2)(2st − 5s − 5t + 15)
d1 d2 d3
(8.14)
and
ω=
2ǫds ∧ dt
dx ∧ dy
=
,
(s − t)2 + 4(s + t) − 21
y(3x − y)(4x + y)
where
d1 = −3 − 12ǫx + 15ǫ2 x2 − 3ǫy − 17ǫ2 xy + 4ǫ2 y 2
d2 = −3 + 12ǫx + 15ǫ2 x2 + 3ǫy − 17ǫ2 xy + 4ǫ2 y 2
d3 = −3 + 27ǫ2 x2 + 10ǫ2 xy + 10ǫ2 y 2 .
The denominator of K ′ is related to K of (8.13) as
d1 d2 d3 = 160ǫ6 (numerator of K) − 27(denominator of K).
72
(8.15)
9
Linearizable mappings
In this section we are going to discuss about linearisable mappings. Roughly
specaking linearisable means that exists a nonlinear transformation of the
dependent variable which bring down the mapping to a linear equation. The
main problem is that such a transformation is complicated and it may have
many steps. So one can wonder if the singularity analysis can be implemented
here. The bad news is that linearisable systems possesses nonconfined singularities so in principle one has to perform an infinite number of blow ups
However here we demonstrate that our method works well also for linearizable mappings. The first example is a simple non-autonomous linearizable
mapping studied in [72]. We show our method is different from that paper
and [63]. The second example is also a linearizable mapping proposed again
by [55] as a discretization of the Suslov system.
9.1
A non-autonomous linearizable mapping
Here we consider the following very simple mapping
x¯ = y
,
y¯ = − xy + an y
(9.1)
where an is an arbitrary sequence of complex numbers. This dynamical system is a linearizable mapping studied in [72] and the degree of this dynamical
system grows linearly and it is lifted to an analytically stable mapping by
blowing up at the following points:
E1 : (x, y) = (0, 0), E2 : (∞, ∞).
The topological singularity patterns are
x
1 x
( , y) = (0, 0) → Hx − E1 → Hy − E2 → ( , ) = (0, 0)
y
x y
(point on E2 ) → Hx − E2 → (curve)
(curve) → Hy − E1 → (point on E2 ).
These are not confined at all. Moreover, we can compute the action on the
Picard group as
¯ x = 2 H x + H y − E1 − E 2
H
¯ y = Hx , E
¯ 1 = Hx , E
¯ 2 = Hx .
H
73
However, since the dynamical system is not an isomorphism, we need to
compute very carefully for this result. One can see detail of such computation
in [72]. Anyway here we are going to linearize the dynamical system using
singularity patterns instead of the action on the Picard group.
From the singularity pattern, we can blow down the surface along Hx − E1 ,
keeping analytical stability. Then we can easily find a basis of blow-down
structure as
H u = H x , H v = H x + H y − E1 − E 2 , F 1 = H x − E1 , F 2 = H x − E 2 .
where the linear systems of Hu and Hv are
| Hu | : u0 x + u1 = 0,
| Hv | : v0 x + v1 y = 0.
Taking new variables u and v as u = x and v = y/x, we have
u¯ = uv
.
v¯ = v + an
9.2
(9.2)
Discrete Suslov system
The discrete Suslov system proposed in [55] is a linearizable mapping:
x¯ − x = ǫa(¯
xy + x¯
y)
.
(9.3)
y¯ − y = −2ǫx¯
x
Again, the degree of this dynamical system grows linearly and it is lifted to
an analytically stable mapping by blowing up at the following points: (we
put a = −b2 for simplicity)
1 1
1 1
E1 : (x, y) = − , 2 , E2 :
,
,
bǫ b ǫ
bǫ b2 ǫ
1
1
1
1
E3 : − , − 2 , E4 :
, E5 : (∞, ∞).
,−
bǫ b ǫ
bǫ b2 ǫ
The topological singularity patterns are
1
x = ∞ → (0, − 2 )
bǫ
y=∞→y=∞
(2bex + b2 ǫy + 1 = 0) → E3 → E2 → (2bǫx − b2 ǫy + 1 = 0)
(−2bǫx + b2 ǫy + 1 = 0) → E4 → E1 → (−2bǫx − b2 ǫy + 1 = 0)
(2b2 ǫ2 x2 − b2 ǫy − 1 = 0) → E5 → (2b2 ǫ2 x2 + b2 ǫy − 1 = 0),
74
where divisor classes are
x = ∞ : H x − E5
x = ∞ : H y − E5
2bǫx − b2 ǫy + 1 = 0 : Hx + Hy − E4 − E5
− 2bǫx − b2 ǫy + 1 = 0 : Hx + Hy − E3 − E5
2b2 ǫ2 x2 + b2 ǫy − 1 = 0 : 2 Hx + Hy − E3 − E4 − E5 .
At first, we blow down along Hx − E5 and Hy − E5 . For that purpose we
take the blow-down structure as
Hs := Hx + Hy − E1 − E5 , Ht := Hx + Hy − E2 − E5 ,
H x − E5 , H y − E5 , H x + H y − E 1 − E2 − E 5 , E3 , E4 .
Then we have a surface whose Picard group is generated by Hs , Ht , E3 ,
E4 where the dynamical system is still analytically stable. We abbreviate
detail, but again we find effective (-1) divisor classes Hs − E3 and Hs − E4 in
singularity pattern which can be blown down preserving analytical stability.
Hence we take a basis of blow-down structure as
Hu := Hs + Ht − E3 − E4 = 2 Hx +2 Hy − E1 − E2 − E3 − E4 −2 E5 ,
u0 (x2 − b2 y 2 ) + u1 (1 − b2 ǫ2 x2 ) = 0,
Hv := Hs = Hx + Hy − E1 − E5 : v0 (1 + bǫx) + v1 (x + by) = 0
H s − E 3 = H x + H y − E1 − E3 − E 5
H s − E 4 = H x + H y − E1 − E4 − E 5 .
If we take the new variables u and v as
u=
x2 − b2 y 2
,
1 − b 2 ǫ2 x 2
then the dynamical system becomes

 u¯ = u
 v¯ =
v=
1 + bǫx
,
x + by
bǫ + v .
1 − bǫuv
75
(9.4)
Remark 9.1. The action of the mapping on the Picard group on the first
surface is given by
¯ x = 2 H x + H y − E 3 − E4 − E 5
H
¯ y = 2 Hx +2 Hy − E3 − E4 −2 E5
H
¯ 1 = 2 H x + H y − E3 − E5
E
¯ 2 = 2 H x + H y − E4 − E5
E
¯ 3 = E2 , E
¯ 4 = E1
E
¯ 5 = 2 H x + H y − E3 − E4 − E 5
E
and Hu is the invariant divisor class whose self-intersection is zero.
A finer classification may be done by the types of singular fibers and the
automorphism of surfaces. Indeed, the symmetries of generalized Halphen
surfaces have a close relationship with the Mordell-Weil lattice of rational
surfaces. However, there are too many types of surfaces and we gave a coarse
but useful classification in this paper.
9.3
Other new linearisable systems
In this section we shall analyse new types of linearisable mappings. Their
forms are inspired by the canonical forms of the QRT mapping twisted in
the logic of replacing products in QRT with ratios. Because the algebraic
geometry here is rather difficult we shall implement the euristic arguments of
degree growth and impose also the same growth for deautonomisation. Also
many of the mappings have transcendental invariant which is not clear how
to extract from the structure of singularities. Practically all o the examples
below will be linearised using the so called Gambier mappings which are
coupled discrete Riccatti equations.
We start with
xn+1 + xn
x2 + axn + b
= f 2n
(9.5)
xn−1 + xn
xn + cxn + d
The investigation of the integrability of (9.5) is carried out using the algebraic
entropy criterion, since we expect some integrable subcases to be linearisable.
We shall not present here the details of this analysis but just the end result.
We find that the only integrable case corresponds to f = 1, c = −a and
d = b. Its degree growth is 1, 2, 3, 4, 5,. . . and thus we expect the mapping
76
to be linearisable. Indeed by considering the Gambier mapping
yn+1 = yn + a
xn =
b + yn xn−1
a − yn + xn−1
(9.6)
(9.7)
and eliminating y we recover the linearisable form of (9.5)
xn+1 + xn
x2 + axn + b
= 2n
xn−1 + xn
xn − axn + b
(9.8)
The mapping (9.8) possesses a transcendental conserved quantity. Indeed,
from the solution of (9.6) we have that yn = na + y0 and thus tan(πyn /a) =
cnst. Solving (9.7) for y we find thus
π xn xn−1 + axn − b
tan
=K
(9.9)
a
xn + xn−1
As a consequence of the linearisability some of the parameters of (9.5)
may be functions of the independent variable. We are thus led to examine
(9.5) afresh, keeping f = 1 but allowing for some less stringent constraint on
a, b, c, d. We require that the degree growth be the same as in the autonomous
case. We find now that the constraints on the parameters are dn = bn−1 and
cn = −an−1 . In order to linearise the mapping we consider now the Gambier
mapping
yn+1 = yn
(9.10)
xn =
and eliminating y we find
bn−1 + (yn − gn )xn−1
gn−1 − yn + xn−1
xn+1 + xn
x2 + (gn − gn+1 )xn + bn
= 2n
xn−1 + xn
xn + (gn − gn−1 )xn + bn−1
(9.11)
(9.12)
where we have introduced the auxiliary variable g through an = gn − gn+1 .
The case where the polynomials in the numerator and denominator of the
rhs of (9.5) are linear is also interesting. We start from
xn+1 + xn
xn + a
=c
xn−1 + xn
xn + b
77
(4.7)
The application of the algebraic entropy integrability criterion leads to c free
while b = −a, and the degree growth is the same a for (9.5). The extension
to a non autonomous case is straightforward: a and c are free functions of
the independent variable n. Thus the linearisable form of the mapping is
xn+1 + xn
x n + an
=c
xn−1 + xn
xn − an−1
(9.13)
The linearisation of (9.13) is given by the Gambier mapping
yn+1 = yn + gn+1
(9.14)
gn an−1 + yn xn−1
(9.15)
g n − yn
Elimination of y leads to (9.13) with cn = −gn+1 /gn . It is interesting to point
out here that even in the autonomous case of constant c the corresponding
Gambier mapping is explicitly nonautonomous since in that case we have
gn = g0 (−c)n . We should also remark that the linearisable case (9.13) can
be obtained from (9.12) by taking x → 0 and an appropriate redefinition of
the auxiliary variables.
Next we analyse the mapping
xn =
xn+1 xn − 1
x2 + axn + b
= f n2
xn xn−1 − 1
xn + cxn + d
(9.16)
Again we start by the purely autonomous case. We find that one linearisable
case exists of the form
x2 + axn + 1/λ
xn+1 xn − 1
= λ2 n2
xn xn−1 − 1
xn + aλxn + λ
(9.17)
Its linearisation is given by the Gambier mapping
yn+1 = yn /λ
(9.18)
xn−1 + yn + a
(9.19)
λyn xn−1 − 1
At this point it is interesting to exhibit a case where (9.17) possesses a
conserved quantity. If we take λ as a root of unity, say λp = 1, then from
p
(9.18) we have yn+1
= ynp . Solving (9.19) for y we have
p
xn−1 + a + axn /λ
=K
(9.20)
xn xn−1 − 1
xn = λ
78
Since p may be any integer we have here an invariant of arbitrary degree.
In order to proceed to the deautonomisation it is preferable to start with
the full freedom of (9.16). We find again that the mapping is integrable in
one linearisable case which has the form
xn+1 xn − 1
bn+1 x2n + an xn + bn
=
xn xn−1 − 1
bn−1 x2n + an−1 xn + bn
(9.21)
Its linearisation is given by the Gambier mapping
yn+1 = yn
(9.22)
bn xn−1 + yn + an−1
(9.23)
yn xn−1 − bn−1
Next we turn to the case where the right hand side of (9.16) is not a ratio of
quadratic but rather of linear polynomials. Two cases can be distinguished
here. The first correspond to a degenerate case of (9.21) where the numerator
and denominator have one common factor. This happens whenever a and b
satisfy the constraint
xn =
(an − an−1 )(an−1 bn+1 − an bn−1 ) − bn (bn+1 − bn−1 )2 = 0
(9.24)
in which case (9.21) degenerates to
bn+1 (an − an−1 )xn + bn (bn+1 − bn−1 )
xn+1 xn − 1
=
xn xn−1 − 1
bn−1 (an − an−1 )xn + bn (bn+1 − bn−1 )
(9.25)
The autonomous limit of (9.25) can be easily obtained. We find that in this
case the constraint is just a = ±(1 + λ) and the mapping becomes
xn+1 xn − 1
1 ± xn λ
=
xn xn−1 − 1
1 ± xn /λ
(9.26)
However a second integrable case does exist which cannot be obtained from
the quadratic one through some limiting procedure. It has the autonomous
form
1 − axn
xn+1 xn − 1
=
(9.27)
xn xn−1 − 1
1 + axn
The degree growth of the iterates of (4.19) is again linear, 1, 2, 2, 3, 3, 4, 4,
5, 5, . . . , an indication that this mapping should be linearisable. This turns
to be the case since (4.19) is equivalent to the Gambier mapping
yn+1 + yn = 0
79
(9.28)
xn =
a + yn + xn−1
1 + axn−1
(9.29)
The deautonomisation of (9.27) is straightforward. We find
xn+1 xn − 1
1 − an x n
=
xn xn−1 − 1
1 + an+1 xn
(9.30)
where an is a free function of the independent variable. The associated
Gambier mapping is exactly (??) where a is now the function an and not
simply a constant.
9.4
Linearisable mappings of Q4 family
In this section we are going to study the mappings given by tavelling wave
reduction of (7.2) and (7.3). It is indeed our experience that when a mapping
is linearisable its coefficients after deautonomisation can be expressed in terms
of some completely arbitrary function (we do not have a poof of this fact,
rather we have observed this in practically all examples we did). This is
indeed the case for projective mappings as well as for the Gambier one. We
are going to work with a general mapping which generalises the reduction of
(7.2)
axn+1 xn−1 + b(xn+1 + xn−1 )xn + cx2n = 1
(9.31)
i.e. a form similar to travelling wave of (7.2) but where the relative coefficient
of the xn+1 xn−1 and x2n terms is not 1 any more. The parameters a, b, c are
now functions of the independent variable.
We are not going to go into all the details of the derivation. It suffices to
say that the linearisation can be obtained in terms of a Gambier mapping.
We subtract (9.31) from its upshift (i.e. taking its discrete derivative) and
reduce the order of the remaining homogeneous mapping by introducing the
auxiliary variable yn = xn+1 /xn . We find the mapping
bn+1 yn2 yn+1 yn−1 +cn+1 yn2 yn−1 +an+1 yn yn+1 yn−1 +(bn+1 −bn )yn yn−1 −an yn −cn yn−1 −bn = 0
(9.32)
This mapping is again a Gambier one. Indeed it can be written as a system
of two discrete Riccati in cascade
yn =
α + zn (β + yn−1 )
yn−1
80
(9.33)
zn+1 = −δ −
zn
γ + κzn
(9.34)
where α, β, γ, δ and κ are functions of the independent variable. In order
to simplify the presentation of the results we introduce the (free) function
gn = bn /an . A detailed calculation shows that it is possible to express the
parameters of the Gambier mapping as follows
αn =
gn−1
gn+1
βn = gn−1
γn =
κn =
gn+1
δn =
1 + gn gn+1
gn−1 bn+1
bn g n
bn+1 gn gn−1
bn gn+1
gn + gn+2
bn gn+1 + gn−1
−
gn+2
bn+1
gn−1
Moreover the three functions a, b and c can be expressed in terms of the free
function g. From the definition of g we have
an =
bn
gn
(9.35)
and moreover we find
cn = g n
bn gn−1 gn−2 (gn+1 + gn−1 ) + bn−1 gn+1 (1 − gn−1 gn−2 )
gn+1 gn−1 gn−2 (1 + gn gn−1 )
(9.36)
while b is given by the linear equation
2
2
bn+1 gn−1 gn−2 (gn−1 gn + 1)(gn+1 gn+2 − 1) + bn gn−2 gn+2 (gn−1
− gn+1
)
+bn−1 gn+1 gn+2 (gn+1 gn + 1)(1 − gn−1 gn−2 ) = 0
(9.37)
Thus equation (9.31) is linearisable and as expected its general nonautonomous
form does involve a free function.
Before concluding this section it would be interesting, as an aside, to
consider the degeneration of the mapping (9.31). As already shown by Adler,
Bobenko and Suris the integrable lattice Q3 does, under the appropriate
81
limiting procedure, degenerate to the lattice these authors of have dubbed
Q2 . In [48] the following reduced form is presented:
(xn+1 − xn )(xn − xn−1 ) + α(xn+1 + 2xn + xn−1 ) + β = 0
(9.38)
and have shown that it is linearisable in the same way as the mapping obtained from the reduction of Q3 . It would be interesting to present here its
deautonomisation. For the linearisation of the autonomous form of (9.38)
we had started by subtracting it from its upshift and reducing the order of
the remaining mapping by introducing the auxiliary variable yn = xn+1 − xn .
Here we start by consider the Gambier mapping:
yn = yn−1 zn + gn (zn + 1)
(9.39)
fn
(9.40)
fn+1
Eliminating z and introducing the variable x we obtain a mapping which can
be written as fn+1 Mn+1 − fn Mn , where Mn = 0 defines a mapping which is
the nonautonomous form of (9.38). We find that f can be explicitly given in
terms of the free function g:
zn+1 zn =
fn =
κgn + 2k(−1)n
(gn + gn−1 )(gn + gn+1 )
(9.41)
where κ and k are two arbitrary constants. The mapping M has now the
form
(xn+1 −xn )(xn −xn−1 )+xn+1 gn−1 +xn (gn −gn+1 +γn (gn +gn−1 ))+xn−1 gn+1 +βn = 0
(9.42)
where γn = (κgn+1 − 2k(−1)n )/(κgn + 2k(−1)n ), βn = −gn−1 gn+1 + (c +
k(−1)n )/fn and c is another free constant. It is clear from the expression
of (9.42) that this nonautonomous form could not have been obtained by
simply allowing the parameters α and β in (9.38) to depend on n.
So one can say that the case of Q2 mapping is more challenging: its
non-autonomous form was obtained from the appropriate limit of the (nonautonomous form of the) Q3 mapping. In this case the straightforward deautonomisation, i.e. allowing the parameters of the mapping to depend on the
independent variable, would not have given the desired result. This should
be an indication for future deautonomisation investigations: in some cases
one must extend the autonomous form, introducing a priori superfluous parameters, in order to ensure a parametrisation rich enough, to be amenable
to deautonomisation.
82
10
Ultradiscrete (tropical) mappings
What is an ultradiscrete system? The name ultradiscrete is used to designate
systems where the dependent variables as well as the independent ones assume only discrete values. In this respect ultradiscrete systems are nothing
but generalised cellular automata. The idea of ‘ultradiscretisation’ comes
from the following question which is crucial for any analysis of a complex
system; how simple can a nonlinear system be and still be genuinely nonlinear? The nonlinearities which we are accustomed with involving simple
integer powers are not the simplest. It turns out that the simplest nonlinear
function of x one can think is |x|. It is linear for both x > 0 and x < 0
but the nonlinearity comes from different determinations. Accordingly any
equation involving nonlinearities only in terms of absolute values will be the
simplest. In fact it will be an equation which is piecewise linear. The ultradiscrete limit converts a nonlinear discrete equation into one where only
absolute value nonlinearities appear. Of course now the dynamics will be
simpler but retains the ‘nonlinear skeleton’ of the initial discrete one. The
only drawback is the positivity requirement for any dependent variable and
parameters.
In order to obtain the ultradiscrete limit we start with an equation for
x, introduce X through x = eX/ǫ and then take appropriate limit ǫ → 0+ .
Clearly the substitution x = eX/ǫ requires x to be positive. The key relation
is:
X + |X|
lim+ ǫ ln(1 + eX/ǫ ) = max(0, X) =
ǫ→0
2
which can be easily generalised to the following basic formulas for sums and
products:
N
X
lim+ ǫ ln(
eXj /ǫ ) = max(X1 , X2 , ..., XN )
(10.1)
ǫ→0
j=1
lim ǫ ln(
ǫ→0+
N
Y
eXj /ǫ ) = X1 + X2 + ... + XN
(10.2)
j=1
In physics this procedure has been applied for the first time in 1996 [75]
in the case of soliton equations and it was shown that indeed the ultradiscrete soliton equations posses multisoliton solution and they behave as in
the discrete case. Also other properties appeared, which are specific to ultradiscrete framework and these are related to the problem of integrability.
83
Mathematically ultradiscretisation procedure is older and it appears for the
first time in computer science. Since then it was developed up to now into full
fleshed topic called tropical mathematics. In order to have a more accurate
understanding we shall define the things more rigurously. We follow the book
[99]. Calling Rmax = R ∪{−∞} we introduce the semiring {Rmax , ⊕, ⊗, ε, e}
through the following definitions:
• a ⊕ b := max(a, b),
• ε := −∞,
a ⊗ b := a + b
e := 0
The following properties are easily verified:
• x ⊕ (y ⊕ z) = (x ⊕ y) ⊕ z and x ⊗ (y ⊗ z) = (x ⊗ y) ⊗ z, ∀, x, y, z ∈ Rmax
• commutativity: (trivial)
• distributivity x ⊗ (y ⊕ z) = x ⊗ y ⊕ x ⊗ z
• zero elements: x ⊕ ε = ε ⊕ x = x, x ⊗ e = e ⊗ x = x
• multiplicative inverse: if x 6= ε,
∃!y, x ⊗ y = e
• absorbing element: x ⊗ ε = εx = ε
• idempotency: x ⊕ x ⊕ x... ⊕ x = x and in general (x ⊕ y)n = xn ⊕ y n
In addition we have the following proposition:
Proposition 10.1. For any a ∈ {Rmax , ⊗, ⊕} with a 6= ε there is no additive
inverse for it
Proof. Suppose a ∈ Rmax has an additive inverse b and a ⊕ b = ε. Then a ⊕
(a⊕b) = a⊕ε = a = (a⊕a)⊕b |{z}
=
a⊕b = ε so a = ε contradiction!
idempotency
Instead of the cumbersome notations ⊗, ⊕ one can use the usual addition
and multiplication signs although it may induce confusions. For instance the
tropical polinomial −2x3 − x2 + x + 5 is max(3x − 2, 2x − 1, x + 1, 5). Also
we have the following definition:
Definition: the tropical (ultradiscrete) hypersurface V (F ) defined by the
tropical polinomial F in n-variables is the nondiferentiable locus in Rn . We
84
will see next section that this locus is essential for defining singularity confinement.
We have seen at the beginning that one can tropicalize any polynomial
(with no minus sign) using the limiting procedures given by the logarithms.
However one can implement a more general procedure using valuation of a
∗
∗
field
P K , νval : K±1 → R.±1 More precisely for any Laurent polynomial f =
ν∈I cν x ∈ K[x1 , ..., xn ] its ultradiscrete version is given by
trop(f )(z) = max(val(cν ) + zν)
. In the next part we are going to use only real valuation for positive numbers
and we implement the usual notation with max.
Remark 10.1. In [99] the tropicalization is taken using the min function instead of max. However things are equivalent since min(a, b) = −max(−a, −b)
It is thus natural at this point to ask how the integrability-related properties of discrete systems carry over to cellular automata obtained from discrete
systems following the ultradiscretisation procedure. The ultradiscretisation
procedure preserves any integrable character of the initial system. One would
thus naturally expect the ultradiscrete analogue of integrability-related properties, like the singularity confinement of the discrete case, to exist. This
would allow one to formulate ultradiscrete integrability conjectures and propose integrability detectors. This question has been already addressed by
Joshi and Lafortune [76] who proposed a singularity analysis approach which
is perceived as the ultradiscrete equivalent of singularity confinement. In this
chapter we shall critically examine this approach and show that the situation
is more complicated than what one would initially expect. In particular we
shall show that, just as in the discrete case, there exist integrable ultradiscrete systems with unconfined singularities but also nonintegrable systems
with confined singularities.
10.1
Ultradiscrete singularities and their confinement
Before proceeding to the analysis of ultradiscrete systems let us recall the
notion of singularity. Given a mapping of the form xn+1 = f (xn , xn−1 ) we
∂xn+1
are in the presence of a singularity whenever ∂x
= 0 i.e., xn+1 “loses”
n−1
its dependence on xn−1 . When this is due to a particular choice of initial
conditions we are referring to this singularity as a movable one. Movable
85
singularities may be bad, for integrability, because they may lead, after a few
mapping iterations, to an indeterminate form (0/0, ∞ − ∞, . . . ) or propagate
indefinitely. In the former case, provided we can lift the indeterminacy while
recovering the lost degree of freedom (using an argument of continuity with
respect to the initial conditions), we are talking about a confined singularity.
As explained in the introduction, mappings which are integrable through
spectral methods have confined singularities. The typical singularity pattern
in this case is the following: the solution is regular for all values of the index
n up to some value ns , then a singularity appears and propagates up to nc
whereupon it disappears and the solution is again regular for all values of
the index larger than nc . In some cases we are in presence of the reciprocal
situation. The solution is singular for all values of n < ns , becomes regular
between ns and nc and is again singular for n > nc . This singularity is
called weakly confined by Takenawa [77] and is considered to be compatible
with integrability. At the limit where there exists no interval where the
solution may be regular, and the solution is singular throughout, we are in
the presence of what we call a “fixed” singularity (which again does not
hinder integrability).
How can these notions be transposed to the ultradiscrete setting? This
is a question that has been addressed by Joshi and Lafortune [76] who proposed an analogue to the singularity confinement property for ultradiscrete
mappings. In the ultradiscrete systems the nonlinearity is mediated by
terms involving the max operator. Typically one is in presence of terms
like max(Xn , 0). When, depending on the initial conditions, the value of Xn
crosses zero, the result of the max(Xn , 0) operation becomes discontinuous:
when X is slightly smaller than 0 the result is zero, while for X > 0 the result
is X. It is this discontinuity that plays the role of the singularity. Typically
if we put X = ǫ, a term µ = max(ǫ, 0) propagates with the iterations of
the mapping and perpetuates the discontinuity unless by some coincidence
it disappears. This disappearance is the equivalent of the singularity confinement for ultradiscrete systems. Joshi and Lafortune [76] have introduced
an algorithmic method for testing the confinement property for ultradiscrete
systems, linked it to integrability and reproduced results on ultradiscrete
Painlev´e equations by initially deautonomising ultradiscrete mappings.
Before proceeding to a critical analysis of the method let us give an illustrative example. In [78] there were introduced three different forms for the
86
ultradiscrete Painlev´e I equations starting from the QRT mapping
xn+1 xn−1 = a
1 + xn
xσn
σ = 0, 1, 2
(10.3)
and its nonautonomous form. In order to illustrate the singularity analysis
approach we shall limit ourselves to the autonomous case and moreover take
σ = 2. Ultradiscretising (10.3) (putting x = eX/δ , a = eA/δ and taking δ → 0)
we find
Xn+1 + Xn−1 = A + max(0, Xn ) − 2Xn
(2.2)
The singularity corresponds to the discontinuity induced by the term max(0, Xn )
when the value of Xn crosses zero. We shall thus examine the behaviour of
a singularity appearing at, say, n = 1 where X1 = ǫ, while X0 is regular and
look at the propagation of this singularity both forwards and backwards. In
what follows we introduce the notation µ ≡ max(ǫ, 0) and the presence of µ
indicates that the value of X is singular. Below we present only the results
corresponding to A > 0, those corresponding to A < 0 leading to similar
conclusions. First we examine the case X0 < 0 and |X0 | < A where one can
see a regular zone between X−3 and X1 and a singular pattern from X2 on
as well as until X−4 .
..
.
X−13 = X−7 − 2X−5
X−12 = X−6 − 2X−5
X−11 = X−5
X−10 = X−7 − X−5
X−9 = X−6 − X−5
X−8 = X−5
X−7 = A + ǫ
X−6 = −X0 − 2ǫ + µ
X−5 = X0 + ǫ − µ
X−4 = A − X0 − ǫ + µ
X−3 = −ǫ
X−2 = X0 + ǫ
X−1 = A − 2X0 − ǫ
X0
X1 = ǫ
X2 = A − X0 − 2ǫ + µ
X3 = X0 + ǫ − µ
87
X4 = −X0 + µ
X5 = A − ǫ
X6 = X3
X7 = X4 − X3
X8 = X5 + X3
X9 = X3
X10 = X4 − 2X3
X11 = X5 + 2X3
..
.
This is a weakly confined case, in the sense that a (small) regular region
exist surrounded by singular values extending all the way to infinity in both
directions. As we explained already such a behaviour is deemed compatible
with integrability. The cases 0 < X0 < A and X0 < −A lead to similar,
weakly confined, patterns. The last case is X0 > A where the solution is
regular until X1 then singular, confined, between X2 and X4 and regular
from X5 on.
..
.
X−3 = A − ǫ
X−2 = X0 − A + 2ǫ
X−1 = −X0 + A − ǫ
X0 = X0
X1 = ǫ
X2 = A − X0 − 2ǫ + µ
X3 = 2X0 − A + 3ǫ − 2µ
X4 = A − X0 − ǫ + µ
X5 = −ǫ
X6 = X0 + 2ǫ
..
.
Thus in all cases we have either a confined singularity (a central singular
zone with regular behaviour outside) or a weakly confined singularity (a
central regular zone with singular behaviour outside). Both behaviours are
deemed compatible with integrability. The two points which we consider
important in this analysis are that a) one must study all possible sectors of
initial conditions and/or parameters and b) one must consider the possibility
of weakly confined solutions.
88
10.2
Nonintegrable systems with confined singularities
and integrable systems with unconfined singularities
As we explained in the introduction there exist discrete systems which while
being nonintegrable still posses confined singularities (for instance the HietarintaViallet mapping [30]). This discovery has as a consequence that singularity
confinement alone cannot be used a discrete integrability detector. As we
shall show now the same problem appears in an ultradiscrete setting. In [79]
there was found a mapping which did pass the confinement test while having
a positive algebraic entropy
1
(10.4)
xn+1 = xn−1 xn +
xn
The main advantage of this mapping over the examples of [30] is that it is
multiplicative and by choosing the appropriate initial data one can restrict
the solution to positive values. In that case the ultradiscretisation of (10.4)
is straightforward. We find
Xn+1 = Xn−1 + |Xn |
(10.5)
where we have preferred to introduce the absolute value of X instead of its
equivalent max(X, 0) + max(−X, 0). We shall examine the behaviour of a
singularity appearing at, say, n = 1 where X1 = ǫ, while X0 is regular.
We again use the identity µ ≡ max(ǫ, 0) = (|ǫ| + ǫ)/2 and distinguish two
different sectors X0 < 0 and X0 > 0. In the first case (X0 < 0) we find the
sequence
..
.
X−3 = 3X0
X−2 = 2X0 − ǫ
X−1 = X0 + ǫ
X0
X1 = ǫ
X2 = X0 − ǫ + 2µ
X3 = −X0 + 2ǫ − 2µ
X4 = ǫ
X5 = −X0 + ǫ
89
..
.
We can see readily that the singularity, indicated by the presence of µ
is confined (to X2 and X3 only). Turning to the case X0 > 0 we find the
sequence
..
.
X−4 = −X0 + 2µ + ǫ
X−3 = −X0 + 2µ
X−2 = ǫ
X−1 = −X0 + ǫ
X0
X1 = ǫ
X2 = X0 + 2µ − ǫ
X3 = −X0 + 2µ
X4 = 2X0 + 4µ − ǫ
..
.
In this case we are in presence of a weakly confined solution: a regular
part around n = 0 is surrounded by unconfined singularities both for large
positive and large negative n’s. Thus the ultradiscrete mapping (10.4) has
confined singularities and is not integrable. (A stronger indication concerning
this nonintegrability, based on growth properties, rather than the analogy
with the discrete case, will be presented in section 5). In this sense system
(10.4) is an ultradiscrete analogue of the equation discovered by Hietarinta
and Viallet [30].
The converse situation, of a mapping which while integrable does not
possess confined singularities does also exist. As expected an example is to
be sought among linearisable systems. In [79] it was discovered the “multiplicative” linearisable mapping
xn+1
xn + a
=a
xn−1
xn + 1
(10.6)
It is straightfoward to check that the parameter a can be always taken larger
than unity. (Indeed it suffices to reverse the direction of the evolution in
which case a goes to 1/a). We can now ultradiscretise (10.6) to
Xn+1 = Xn−1 + A + max(Xn , A) − max(Xn , 0)
(10.7)
where A > 0. The complete description of the solution would require examining several sectors exist but in order to show that there exist unconfined
90
singularities it suffices to exhibit such a situation in one sector. It turns out
that the case where X0 has a large negative value is one leading to unconfined
singularities.
..
.
X−4 = −X0 − 4A
X−3 = −4A + ǫ
X−2 = X0 − 2A
X−1 = −2A + ǫ
X0
X1 = ǫ
X2 = X0 + 2A − µ
X3 = 2A + ǫ
X4 = X0 + 3A − µ
X5 = 4A + ǫ
X6 = X0 + 4A − µ
X7 = 6A + ǫ
..
.
We remark readily that while for negative indices the solution is regular,
a singularity, mediated by µ, appears for positive n’s and is never confined.
We will analyse mapping (10.7) from the point of view of the growth of the
solutions as well.
Thus in perfect parallel to the discrete situation there exist ultradiscrete
systems where despite the nointegrable character we have confined singularities while for ultradiscrete systems obtained from linearisable mappings the
singularities are not confined.
10.3
A family of integrable mappings and their ultradiscrete counterparts
In this section we shall pursue the study of the singularities of ultradiscrete
systems which come as limits of mappings of the QRT family and discuss
their special properties. In particular we shall examine a mapping of the
form:
(xn+1 xn − 1)(xn xn−1 − 1) =
x4n + ax2n + 1
(1 + xn /b)σ
91
σ = 0, 1, 2
(10.8)
Mapping (10.8) is a special subcase of the autonomous limit of q-discrete
Painlev´e V. When σ = 0 the mapping was shown in [80] to be linearisable.
All three cases belong to the QRT family and do possess a conserved quantity.
We introduce yn = xn+1 xn − 1 and (with obvious notations) we obtain the
ultradiscrete form of (10.8)
Xn+1 = −Xn + max(Yn , 0)
(4.2)
Yn = −Yn−1 + max(4Xn , 2Xn + A, 0) − σ max(Xn − B, 0)
Let us concentrate first on the σ = 0 case. The singularity corresponds here
to the value of Y crossing 0. We thus put Y0 = ǫ and iterate (??) starting from
X0 both backwards and forwards. We examine the branch 0 < X0 < A/2.
This is the sequence we find for n < 0
Xn = X0 + n(A − ǫ)
Yn = Xn + Xn+1
(10.9)
At n = 0 we have by definition X0 and Y0 = ǫ. At n = 1 we find a singular
value
X1 = −X0 + µ
(10.10)
and iterating for positive n we obtain
Xn+1 = X1 + n(A − ǫ)
Yn = Xn + Xn+1
(10.11)
Since Xn+1 contains X1 , the singularity which appeared at n = 1 propagates
ad infinitum. On the other hand since (10.8) with σ = 0 is a member of the
QRT family it does have an invariant:
K=
x2n + x2n−1 + a
xn xn−1 − 1
(10.12)
Ultradiscretising (10.12) is straightforward
K = max(4X, 2X + A, 2 max(Y, 0)) − 2X − Y
(10.13)
We can check that (10.13) is indeed conserved by (10.8) and at no point does
the singularity hinder this conservation.
92
Thus we are here in the presence of an integrable mapping with unconfined
singularities. This counterexample to the integrability criterion of [76] is even
more serious than the examples of the pevious subsection since the mapping
here possesses an explicit invariant. It is thus natural to wonder what does
happen in the remaining cases of (??), σ = 1 and 2. Presenting exhaustive
results, as in the case of section 2, would be prohibitively long. Below we
present a few typical numerical examples. We start with the case σ = 2, take
parameters A = 100 and B = 11, and initial condition X0 = 7. We obtain
the sequence:
..
.
X−3 = −15 + ǫ − µ
Y−3 = ǫ
X−2 = 15
Y−2 = 122
X−1 = 107
Y−1 = 114
X0 = X0
Y0 = ǫ
X1 = −7 + µ
Y1 = 86 − ǫ + 2µ
X2 = 93 − ǫ + µ
Y2 = 122 − ǫ
X3 = 29 − ǫ + µ
Y3 = ǫ
X4 = −29 + 2µ
Y4 = 42 + 3ǫ − 4µ
..
.
We remark that this is a weakly confined singularity. A regular pattern
exists between Y−3 and Y0 and the singularity extends all the way to ±∞ on
the outside. What is more interesting is that the value of Y comes backs to
zero, up to a quantity of O(ǫ), repeatedly albeit not in a periodic way. As a
matter of fact the values of n for which Y is of order ǫ do show some regularity:
. . . ,-26, -22, -19, -16, -13, -10, -6, -3, 0, 3, 7, 10, 13, 16, 19, 23, 26. . . . We
remark that the interval between two successive zeros is either 3 or 4 but as
far as we can tell there is no particular regularity in the succession of these two
numbers. Similar results can be obtained in the σ = 1 case. Again we find
a weakly confined singularity and the zeros of Y appear at values: . . . , -34,
93
-29, -25, -21, -17, -13, -8, -4, 0, 4, 9, 13, 17, 21, 25, 30, 34. . . . By studying
the variation of the (mean) length of the intervals between two successive
zeros of Y , which, we point out again here, give also the length of the regular
zone, we arrive at the following conclusion. For fixed (appropriate) values
of X0 and A and increasing values of B, with 2B/A integer, the length is
exactly 2B/A + 3 for σ = 2 and 2B/A + 4 for σ = 1. If Y0 takes exactly
the value 0 then the solution is strictly periodic. If 2B/A is not integer then
these quantities give the mean length of the interval. We can now see what
is happenning in the σ = 0 case. We can obtain this case by starting from
σ = 1 or 2 and take B → ∞. Thus at the limit the length of the regular zone
becomes infinite and we go from a situation of weakly confined singularities
to one of an unconfined singularity.
At this point one can wonder what is happening in the case where the
mapping is not integrable. We take (??) with σ = 3 and choose the same
parameters as for the case analysed just above, namely A = 100, B = 11 with
initial conditions X0 = 7 and Y0 going through zero. Iterating the mapping
we find that the solution does not recur to O(ǫ) although it does repeatedly
cross zero to change sign. So for negative n the solution is regular while for
positive values of n the singularity continues indefinitely. Thus in this case
we have unsurprisingly an unconfined singularity.
In our analysis above we have presented the “interesting” singularity patterns. There also exist ranges of parameters in combination with the initial
value X0 for which the solution has strictly confined singularities. Their study
does not bring any new element: it suffices that one unconfined singularity
pattern exist for confinement to be violated.
10.4
Complexity growth of ultradiscrete systems
As we have seen in the previous sections the situation concerning the integrability criterion of [76] is far from clear. Counterexamples exist both
as to its sufficient and as to its necessary character. This does not mean
that the criterion is not useful. As was shown by Joshi and Lafortune there
exist many instances where the criterion can be put to use and succesfully
predict integrable deautonomisations. Still, because of the counterexamples,
one is tempted to look for auxiliary or complementary criteria. Since in the
discrete case growth arguments turned out to be crucial for integrability it
makes sense to try to adapt these arguments to the case of ultradiscrete
systems.
94
Clearly the complexity argument used in the case of discrete systems
(and its implementation through the algebraic entropy techniques) cannot
be transposed as such to the ultradiscrete case. Still the growth of the values
of the variable can be of interest as we shall see in what follows.
We start with the integrable ultradiscrete system (??) and iterate it backwards and forwards for parameter A = 7 and initial values X0 = −100 and
X1 = 0. We find the following sequence of values: . . . , -100, 107, 0, -100,
207, -100, 0, 107, -100, 100, 7, -100, 200, -93, -7, 114, -100, 193, -86, -14, 121,
-100, 86, 21, . . . . We remark that the solution does not grow but oscillates
around zero. As a matter of fact the absolute value of the solution never
exceeds the value 2|X0 | + |A|. Similar results can be obtained for other values of the parameter and initial conditions. Another integrable ultradiscrete
system with an explicit conserved quantity is (??). We have given above
numerical values of the iterates of the case σ = 2,with parameters A = 100,
B = 11, and initial condition X0 = 7. Again the solution is not growing but
bouncing between values which in this case never exceed 2B + A.
It may turn out that this property of bounded, bouncing solution is characteristic of a certain class of integrable ultradiscrete systems. Clearly more
detailed studies are needed before one can make a more affirmative statement.
What is clear at this stage is that not all integrable ultradiscrete systems do
have such solutions. Analysing the growth of (??) with σ = 0 (which in the
discrete case is not just QRT-integrable but in fact linearisable) we find the
sequence of values, for A = 100 and X0 = 7, Y0 = 0. We have for X: . . . ,
207, 107, 7, -7, 93, 193, 293. . . and values that grow linearly by steps of 100
away from zero in both positive and negative directions. Similarly for Y we
find: . . . , 314, 114, 0, 86, 286, . . . and linear growth in steps of 200 away from
zero in both directions. In order to investigate whether this linear growth
is a property of ultradiscrete systems coming from linearisable mappings we
analyse the solutions of (10.7), taking A = 10, X0 = 0 and X1 = 7. We
find the sequence: . . . -60, -53, -40, -33, -20, -16, 0, 7, 13, 17, 23, 27, 33, 37,
. . . . Again we have a linear growth of the solution. Towards negative n the
solution grows with alternating steps of 7 and 13 while for positive n we have
alternating steps of 4 and 6. Another example can be given by the mapping
Xn+1 = −Xn−1 + Xn + max(Xn , 0)
(10.14)
which comes from the linearisable discrete system xn+1 xn−1 = xn (xn + 1).
Again starting from initial conditions X0 = 0 and X1 = 1 we find Xn = n,
obvisouly a linear growth.
95
While integrable mappings have moderate growth nonintegrable ones like
(3.2) may grow much faster. By inspection we conclude that the solutions of
(10.5) form a Fibonacci sequence and thus grow exponentially fast. On the
other hand exponential growth is not the only possible one. For instance if
we consider the ultradiscrete analogue of (10.3) with σ = −1, which is not
integrable, we find that the growth of the solutions is quadratic. What is
making the situation even more complicated is for (??) with σ = 3, which is
clearly a nonintegrable case, we find a bounded, bouncing solution.
In view of the above here are the (few) conclusions one can draw with
respect to growth properties of ultradiscrete systems. If one finds an exponential growth of the values of the iterates this is an indication of nonintegrability, while a linear growth indicates linearisability. However one must
bear in mind the fact that even in these cases a slower growth may be possible. Thus the growth properties for ultradiscrete systems can be of some
assistance in the detection of integrability but they do not constitute a powerful tool as in the discrete case. Probably a setting up in terms of tropical
algebraic geometry would be more helpful.
10.5
Linearisable ultradiscrete dynamics: example from
a biological model
In this section we discuss the tropicalization of a system of partial discrete
equations which is linearisable. This system models a modular genetic network and it was published in [6]. We will show that it supports travelling
wave solutions which exists also at the tropical level. Moreover a new periodic
solution it is shown to exist at the tropical limit.
The model itself has the following form:
α + βp3n−1 (t − τ )
− λp3n (t)
1 + p3n−1 (t − τ )
a + bp3n (t − τ )
− λp3n+1 (t)
p˙3n+1 (t) =
1 + p3n (t − τ )
A + Bp3n+1 (t − τ )
p˙3n+2 (t) =
− λp3n+2 (t)
1 + p3n+1 (t − τ )
p˙3n (t) =
(10.15)
(10.16)
(10.17)
where (α, β), (a, b), (A, B) are the parameters characterising promoters of
the genes in the group. We are going to consider here λ to be high - which
96
happens only in artificially circuits by means of specific peptide sequences
appended to the proteins to make them targets for proteases in the cell.
Because we have nonlinear partial differential discrete equations with delay is more convenient make them fully discrete by writting time derivative
as a finite difference.
α + βp3n−1 (t − τ )
p3n (t + δ) − p3n (t)
=
− λp3n (t)
δ
1 + p3n−1 (t − τ )
p3n+1 (t + δ) − p3n+1 (t)
a + bp3n (t − τ )
=
− λp3n+1 (t)
δ
1 + p3n (t − τ )
p3n+2 (t + δ) − p3n+2 (t)
A + Bp3n+1 (t − τ )
=
− λp3n+2 (t)
δ
1 + p3n+1 (t − τ )
(10.18)
(10.19)
(10.20)
Since λ can be made artificially big we can choose the time step δ to balance
it, namely λ = 1/δ. Taking for conveninence the notations with specific
staggering p3n−1 := zn−1 , p3n := xn , p3n+1 := yn , p3n+2 := zn ... we have the
following tractable form of rate equations:
α + βzn−1 (t)
1 + zn−1 (t)
a + bxn (t)
yn (t + σ) =
1 + xn (t)
A + Byn (t)
zn (t + σ) =
1 + yn (t)
xn (t + σ) =
(10.21)
(10.22)
(10.23)
where σ = τ + δ.
In order to solve the system of equations (10.21), (10.22) and (10.23)
we eliminate yn from (10.22) and (10.23) and then plug into (10.21). The
resulting equation will be:
xn (t + σ) =
µ + νxn−1 (t − 2σ)
ρ + γxn−1 (t − 2σ)
where:
µ = α(1 + a) + β(A + aB)
ν = α(1 + b) + β(A + bB)
ρ = 1 + a + A + aB
γ = 1 + b + A + bB
97
(10.24)
In order to show the modularity of the whole network we consider:
xn (t) →
ρ
xn (t),
γ
αe =
µ
,
γ
βe =
νρ
γ2
With these substitution the above equation is transformed into
xn (t + 3σ) =
αe + βe xn−1 (t)
1 + xn−1 (t)
(10.25)
The equation (10.25) can be immediately linearised by the Cole-Hopf type
transform,
Fn (t + 3σ)
xn (t) = −1 +
Fn−1 (t)
to the following linear equation
Fn (t + 6σ) − (αe − βe )Fn−2 (t) − (1 + βe )γ)Fn−1 (t + 3σ) = 0
(10.26)
We check for travelling wave signals, i.e., Fn (t) = F (ξ) where ξ = n + vt,
and v is the signal velocity. Equation (10.26) becomes
F (ξ + h) − (1 + βe )F (ξ) − (αe − βe )F (ξ − h) = 0
(10.27)
where h = 1 + 3vσ is the step. The speed v is free but the product vσ must
be an integer. The solution can be easily computed and has the following
form:
√ !ξ/h
√ !ξ/h
1 + βe − ∆
1 + βe + ∆
+ C2
(10.28)
F (ξ) = C1
2
2
where ∆ = (1 + βe )2 + 4(αe − βe ) and C1,2 are integration constants. Since
the solution must be positive, both terms in the right-hand-side
of equation
√
(10.28) should be as well. So we must have (1 + βe ) > ∆ which leads to
αe < βe ⇔ (a − b)(α − β)(A − B) < 0. Another condition comes from the
fact that pn (t) = −1 + F F(ξ+h)
must be positive. Now if αe = 0, βe > 1 then
(ξ)
the solution has the following kink-type shape
pn (t) = −1 +
1 + Cβeη+1
βeη
=
C(β
−
1)
e
1 + Cβeη
1 + Cβeη+1
98
(10.29)
where η = (n + vt)/h, C and v are arbitrary. This solution shows that along
the transcriptional modular cascade, we have a successive gene expression, all
genes being sequentially expressed as the signal kink goes on (for biological
relevance see [6], [97]
Now we are going to analyse the ultradiscrete limit of our discrete equations. Since the modular cascade is equivalent with a network having same
gene we shall treat only the equation,
xn (t + 3σ) =
αe + βe xn−1 (t)
.
1 + xn−1 (t)
Of course now the dynamics will be simpler but retains the ‘nonlinear
skeleton’ of the initial discrete one. As we have seen in the previous chapter
the method of ultradiscretisation is algorithmic and extremely simple. We
have applied the ultradiscrete approach to many biological models [85, 86,
87, 88]. The only drawback is the positivity requirement for any dependent
variable and parameters, but here this is not a problem since all the biological
quantities are positive. We are going to apply this method to show the
linearisability and and how kink nonlinear wave survives at ultradiscrete
limit. In addition we shall show that a peiodic solution exists.
In order to obtain the ultradiscrete limit we start with an equation for
x, introduce X through x = eX/ǫ and then take appropriate limit ǫ → 0+ .
Clearly the substitution x = eX/ǫ requires x to be positive.
For our equation we put xn (t) = eXn (t)/ǫ , αe = eAe /ǫ , βe = eBe /ǫ and obtain
finally:
Xn (t + 3σ) = max(Ae , Be + Xn−1 (t)) − max(0, Xn−1 (t))
which can be written in a more convenient form (using the distributivity of
max-operation with respect to addition) as:
Xn (t + 3σ) = max(Ae − Be , Xn−1 (t)) − max(0, Xn−1 (t)) + Be
(10.30)
One can see that if the parameters Ae , Be and initial conditions Xn (0) are
integers then the evolution will produce only integer results, so our equation
is indeed a generalised cellular automaton. In addition, the variable Xn (t) is
no longer positive, since it is related to the logarithm of the initial one xn (t)
In order to discuss the solution we impose the travelling wave ansatz
ν = Kn + Ωt with {n, t, K, Ω} ∈ Z. In this way one obtains a discrete
99
piecewise linear equation in one integer variable ν shifted by the integer
value µ = K + 3σΩ. Calling Se = Ae − Be we have:
X(ν + µ) = max(Se , X(ν)) − max(0, X(ν)) + Be
(10.31)
This equation can be solved by reducing to linear discrete equations of order
µ on various sectors defined by the signs of Se , Be or X. Since µ is free we
have a lot of possible solutions. For simplicity we take µ = 1 and show the
solutions:
• Case Se < 0 and Be < 0; for Se < X < 0 we have X(ν +1) = X(ν)+Be
with the solution X(ν) = Be ν +c1 (the initial condition is related to the
constant c1 ). But Se < X < 0 gives [Ae /Be −1−c1 /Be ] < ν < −[c1 /Be ].
Because Ae /Be > 0 we have 0 ≤ Ae /Be ≤ 1 and accordingly n will
have only one or maximally two values. So the solution is trivial. For
X < Se < 0 we have X(ν + 1) = Se + Be = Ae which gives a constant
solution X(ν) = Ae .
Now for X > 0 the only sector is given by Se < 0 < X, X(ν + 1) = Be
with the constant solution X(ν) = Be > 0 - contradiction. So, for
Se < 0 and Be < 0 we have only constant solution.
• Case Se < 0 and Be > 0. Again for X < Se < 0 we have X(ν + 1) =
Se + Be = Ae giving a constant solution X(ν) = Ae which can be
positive or negative. Also for Se < X < 0 we have X(ν+1) = X(ν)+Be
with the solution X(ν) = Bν + c1 and again we have an interval for
ν as above. But now Be is positive and in this case we can make the
solution to be nontrivial choosing −Ae to be huge namely Ae = −∞
(this is not a problem; the biological parameter αe = 0 in this case).
So indeed X(ν) = Be ν + c1 for all ν < [−c1 /Be ]. Now for X > 0 the we
have X(ν + 1) = Be with the constant solution X(ν) = Be > 0. These
sectors can be unified to give a sigle form of the solution which is not
trivial and has a travelling wave form:
X(ν) = Be (ν + 1) + c1 − max(0, Be ν + c1 )
This solution is nothing but the ultradiscrete limit of the discrete kink
solution (10.29) obtained in the case αe = 0 ⇔ Ae = −∞ and βe >
1 ⇔ Be > 0
100
• all other sectors give trivial solution except the following; 0 ≤ X ≤
Se , Be < 0. We have X(ν + 1) = Ae − X(ν) with the solution
X(ν) =
Ae
+ (−1)ν c2
2
(10.32)
This solution is a periodic one which has meaning only in the case of
integer ν. For an appropriate choice of c2 (for instance c2 = Ae /4) the
solution is smaller than Se and bigger than zero. This solution shows
that we have a signal propagating also in periodic networks.
However, the fact that we have a periodic solution in the ultradiscrete
limit does not guarantee the existence of such solution in the initial discrete
equation. It may correspond not only to a periodic but also to a damped
oscillating solution. Moreover we first linearised our discrete equation by a
Cole-Hopf type transform and then compute solutions. It may happen that
some solutions do not belong to the linearisable sector (not captured by the
Cole-Hopf). As we said at the beginning of the section, the ultradiscrete limit
retains only the skeleton of the initial discrete equation and accordingly not
everything in the discrete case have an unique correspondent in the ultradiscrete one. Also the reverse is possible, for instance multiple ultradiscrete
limit cycles which correspond to only one in the discrete case or negative
ultradiscrete solitons with no counterpart in the discrete case as well [98].
We are going to end this section showing how the linearisability and ColeHopf transform works in the ultradiscrete case. Here we have a problem. The
Cole-Hopf transform involves a negative minus one term
xn (t) = −1 + Fn (t + 3σ)/Fn−1 (t).
In order to eliminate this problem we will rewrite the equation in the variable
wn (t) = 1 + xn (t) as :
wn (t + 3σ) =
αe − βe
+ 1 + βe
wn−1 (t)
(10.33)
The main drawback now is that only αe − βe > 0 is compatible with ultradiscretisation. With the substitution wn (t) = eWn (t)/ǫ , ae − be = eSe /ǫ , 1 + be =
eQ/ǫ , Fn (t) = eΦn (t)/ǫ we have:
Wn (t + 3σ) = max(Se − Wn−1 (t), Q)
101
(10.34)
Now, put the ultradiscrete Cole-Hopf Wn (t) = Φn (t + 3σ) − Φn−1 (t) in
(10.33). The equation goes down to:
Φn (t + 6σ) = max(Se + Φn−2 (t), Q + Φn−1 (t + 3σ))
(10.35)
which is nothing but the ultradiscrete limit of the linear discrete equation
(10.27) Of course the term linearisability is somehow invisible for ultradiscrete
equation inasmuch as they are already piecewise linear. But the equation
(3.25) has an additional symmetry with respect to Φn (t) → Φn (t) + hn (t) for
any function hn (t). This is the way of manifestiation of the linear character
at the ultradiscrete level.
102
11
General conclusions
The main topic we covered in this thesis is the role of singularities in establishinjg the integrable character of a mapping (discrete or ultradiscrete)
and in integrating it effectively. Even though at the begging the criteria
were introduced from the physicist point of view namely pure euristic, gradually it was realised that the rigurous approach based on algebraic geometry
tools can improve tremedously their effectiveness. We tried our best in this
thesis to underline the following aspects whenever one deals with a discrete
mapping: the integrable character can be established either by computing
(numerically) complexity growth, or by analysing rigurously the singularities.
Now for complexity growth,
• if the complexity growth is linear then the mapping is a linearisable
one
• if the complexity growth is quadratic then the mapping is still integrable by means of spectral methods (Lax pairs) and it involves elliptic
functions.
• if the complexity growth is exponential then the system has positive
algebraic entropy which by the theorem of Gromow-Yomdin [92] implies a positive topological entropy i.e. no-integrable one. However
there are systems which escapes from this namely the ones that can
be linearised by non-rational or transcendental transformations. For
instance, xn+1 xn−1 − xpn = 0, p ≥ 3 has nonconfined singularities and
positive algebraic entropy. But by means of zn = log xn the system can
be linearised to zn+1 + zn−1 − pzn = 0. Still the integrability of this linear equation is problematic due to multivaluedness of the logarithnmic
substitution. In addition the chaotic character of the original mapping
is not rigurousluy established beacuse from the numerical experiments
seems rather an ergodic behaviour. Accordingly we do not take into
account these type of systems.
Even though the complexity growth is very effective as an integrability
criterion the main problem is to integrate effectively the mapping. Here the
singularity confinement enters on the stage by giving the pattern of singularities. Blowing up this singularities one obtains (if the number of singularities
is finite) a rational elliptic surface (which must be minimised in case not minimal). If the orthogonal complement of the associated singular fibers Dynkin
103
diagram is a affine Weyl group then the systems is an integrable one. The
invariant can be computed from the proper transform of the divisor class
of eigenvalue one of the action on the associated Picard group (or echivalently Neron-Severi lattice). If the corresponding Weyl group is not affine
then the system is non-integrable (even though realises an automorphism of
an algebraic surface). In addition singularity confinement produces deautonomisation by imposing the same singularity pattern in case of a mapping
with unknown coefficients
The case of linearisable systems is more complicated. However still from
singularity pattern it is possible to find the linearisation and even to deautonomise the system. The most enigmatic domain is the integrability of
tropical (ultradiscrete) mappings. Being piecewise linear one is tempted to
say that they cannot be chaotic at all. However there is no algorithm to
find their invariants. The presence of nondetermiantion points given by the
max function suggested that a kind of confinement can be idone namely disapearance of such nondetermination with initial data recovering. However
we have shown that this criterion although instrumental in establishing integrability for simple QRT-like mappings and lattice soliton equations, fails in
many other cases and what is worst there is no analog of complexity growth.
We expect that a deep understanding of notion of singularity based on the
concepts of toric and tropical algebraic geometry will shed light on the clarification of such problems.
We are ending the section by saying that discrete mappings or discrete
soliton equations can have important applications in molecular biological
models. We have written many papers on this topic and we intend to analyse mathematically various equations coming from quantitative molecular
biology. T
12
Future research directions
As we have seen the instruments of algebraic geometry are very effective in
analysis of the dynamics of two dimensional mappings. Even though many
results are already known we intend to continue this approach to unveil other
beautiful features of integrability and possible applications.
1. Higher order mappings. This is the natural step which we intend to tackle
in the near future. Of course here exists a major drawback. Since now we
have to work on P3 or P1 × P1 × P1 we can have singularities which are not
104
only points but also curves - and this fact will overburden enormously the
computations. By developing certain software techniques (in MATHEMATICA or McCAULEY) we will consider the singularity patterns here and see
how to construct the invariants. Also nothing has been done on the systems
having invariants parametrised by hyperelliptic curves.
2. Ultradiscrete mappings. As we have seen in the last chapter integrability
and invariants represents open problems. Also construction of a tropical QRT
mapping is still problematic despite the results obtained by [91]. We intend
to rely on the properties of tropical elliptic curves and try ”brute force” to
find some extension and improvement of ultradiscrete singularity confinement
presented in the last chapter. Also we intend to study the connection with
toric varieties although because the of the C∗ action, they are not abelian
so it will be difficult to imagine an integrable mapping with toric level sets.
On the other hand we intend to study more carefully the symmetries of the
ultradiscrete Painlev´e equations using the recently introduced ultradiscrete
Hirota bilinear formalism [94] (this approach was used for discrete Painlev´e
equations before the algebraic geonmetric one). In the same direction the
soliton dynamics for partial ultradiscrete equations is still at the beginning
and apart from Korteweg de Vries and Toda systems there are very few
studies on others [95].
3. Connection between geometry and Lax pairs (represented here by isomonodromic deformation). For instance the q-Painleve I equation
xn+1 xn−1 =
zn zn+1 (1 + xn )
,
x2n
zn = αq n/2
has the following Lax pairs

0
0
zn /xn
0
 0
0
xn−1 qxn−1
Ln = 
 λxn
0
1
q
0 λzn−1 /xn−1
0
0
and





0 xn /zn (1 + xn )
0
0
 0
0
1
0 

Mn = 
 0
0
1/xn q/xn 
λ
0
0
0

105
The deformation of the q-linear difference system is given by:
φn (qζ) = Ln (ζ)φn (ζ)
φn+1 (ζ) = Mn (ζ)φn (ζ)
Complatibility of these two equations is:
Mn (qζ)Ln (ζ) − Ln+1 (ζ)Mn (ζ) = xn+1 xn−1 −
zn zn+1 (1 + xn )
x2n
The existence of such compatibility for a deformation of a linear systems
is a fundamental aspect of integrability. For the moment there are very few
studies [93],[?] about connection between the emergent space of initial conditions (geometry of a rational surface) and construction of such operators.
We intend to extend the results of [93] to any type of Halphen surface.
4. Isomonodromic deformations and space for initial conditions for delayequations. Delay mappings are a hybrid between discrete and continuous
systems. For instance the well known delay-Painleve II equation:
d
w(t)w(t + 1) = w2 (t) − w2 (t + 1)
dt
which can be deduced from the travelling wave reduction of the famous soliton equation sine-Hilbert Hut (x, t) = sin u(x, t) (H is the Hilbert transform
with respect to x) . Very few things are known for such systems. There are
only two papers concerning them one by Ramani et al. in 1992 [96] where
the singularity confinement+Painleve test were mixed and Carstea 2010 [90]
where the Hirota bilinear forms were obtained. We intend to study more carefully these systems relying on the fact that their Lax pairs can be obtained
by reduction of Lax pairs of integro-differential soliton equations involving
singular integral operators.
5. Fermionic extensions of lattice soliton equations Although we did not treat
in this thesis the supersymmetric integrability, we do have many significant
results concernig dynamics of supersymmetric solitons. For almost 10 years
the topic has been focusing on continuous systems like supersymmetric KdV
hierarchy and modified KdV and we practically initiate the domain related to
Hirota super-bilinear formalism [81, 82, 83, 84]. But very recently appeared
results concerning fermionic extensions of partial discrete equations. We
intend to study the Hirota bilinear formalism in this context and interaction
106
of fermionic lattice solitons. Even though Painleve analysis can be applied
to such equations (and we also constructed supersymmetric extensions of
Painleve I and II equations) nothing has been done in the discrete context.
We intend at the beginning to understand what means a singularity in lattice
equations with values in Grassmann algebra.
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