Growth, invariants, Lagrangians and integrability for
four-dimensional recurrence relations
Giorgio Gubbiotti
joint work with N. Joshi, D.T. Tran and C-M. Viallet
School of Mathematics and Statistics, The University of Sydney,
New South Wales 2006, Australia
giorgio.gubbiotti@sydney.edu.au

Symmetry and Integrability of Difference Equations 13
Fukuoka, Kyūshū, Japan
November 11–17 2018
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Outline of the Seminar

1

Bi-rational projective maps and integrability

2

Duality3

The classification

4

Results and outlook

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Bi-rational projective maps
In this talk we will deal with recurrence relations.
For us a recurrence relation of order n is a bi-rational map from then-dimensional complex projective space into itself:
ϕ : [x] ∈ CPn → [x0 ] ∈ CPn .

0
are theHere [x] = [x1 : x2 : · · · : xn+1 ] and [x0 ] = x10 : x20 : · · · : xn+1
homogeneous coordinates in projective space and:x10 = P,xi0 = xi−1 Q, i = 2, . . . , n,0xn+1
= xn+1 Q,

where P, Q ∈ Ch [x1 , . . . , xn+1 ].
We are interested in the study of the integrability of such maps.

Bi-rational maps
A rational map of algebraic varieties ϕ : V → W is bi-rational if there exists a
rational map ψ : W → V such that it is the inverse of ϕ in the subset where bothare defined.
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Invariants of bi-rational maps
Integrability for recurrence relations can be understood in different ways.
Like for continuous differential equations, since we are in the autonomouscase we can introduce the concept of invariants.

First integrals
An invariant for a bi-rational map is a homogeneous function I : CPn → C suchthat ϕ∗ (I ) = I . With ϕ∗ (I ) we mean the pullback of I through the map ϕ, i.e.
ϕ∗ (I ) = I (ϕ ([x])).

Degree patternN

Let F ∈ Ch [x1 , . . . , xn+1 ] , i.e. F is a vector of homogeneous polynomials. We
define the degree pattern of F to be:dp F = degx1 F , degx2 F , . . . , degxn F ,where the degxi F is maximum of the degrees of the entries of F with respect to xi .
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Integrability for bi-rational maps
Invariants are useful in solving recurrence relations because the existence ofan invariant I means that the solutions of the recurrence relation must lie on
the hypersurface I = κ.
The value of the constant κ is determined by the initial conditions x0 :
κ = I ([x0 ]).
This means that, in principle, if we know n − 1 functionally independent
invariants of an n-dimensional map ϕ by solving the system
Ii = κi ,i = 1, . . . , n − 1

we can reduce the map to a 1-dimensional map ϕb : CP1 → CP1 .
The solution is then obtained by integrating the map ϕb which will give a lastinvariant.

Integrability
An bi-rational map is a integrable if it possesses n − 1 invariants.

Reduction and bi-rationality
In general the reduction to a lower-dimensional map can break the bi-rationality.
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How many invariants are really needed for integrability?

From the previous discussion we have that for a general n-dimensionalbi-rational map to claim integrability it is necessary to produce n − 1
invariants, and then a last invariant will come from integration.
So to have integrability one needs n invariants, but if the map has some
additional structure the number of integrals can be significantly lowered.
In the continuous case, for even-order differential equations a natural
additional structure to look at is the Lagrangian/Hamiltonian formulation.
In particular in Hamiltonian mechanics a powerful result is the so-called
Liouville theorem which states that for the integrability in the Hamiltoniancase only half of the invariants are needed.
In the map setting there is no commonly accepted equivalent of the
Hamiltonian formulation, but it is still possible to define symplectic structures
for even-dimensional maps.

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Symplectic forms
Symplectic forms and symplectic maps
In affine coordinates y = (y1 , . . . , y2r ) = [x1 /x2r +1 : · · · : x2r /x2r +1 : 1] ∈ C2r the
2-form:X
ωi,j (y) dyi ∧ dyj
ω=i<jis called symplectic form if is closed and non-degenerate a.e. in C2r .

Symplectic maps
A map ϕ : C2r → C2r is called a symplectic map if its pullback preserves the
symplectic form ω: ϕ∗ (ω) = ω.

Poisson bracket
A symplectic form defines a Poisson bracket through the equation{f , g } :=

2r
X
i,j=1

G. Gubbiotti (School Math. & Stat. USYD)Ωi,j (y)∂f ∂g,∂yi ∂yj−1Ω = (ωi,j )Growth, invariants, Lagrangians and integrability.
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Symplectic maps and Liouville theorem
For symplectic maps the following theorem holds:

Discrete Liouville Theorem (Bruschi and Ragnisco 1991, Veselov 1991)
If symplectic map ϕ : C2r → C2r possess r invariants I1 , . . . , Ir , such that
{Ik , Il } ≡ 0 for every k, l = 1, . . . , r then it is possible to find the r − 1 missing
invariants and reduce the maps to a discrete integration. Moreover, if theinvariants define compact and connected hypersurfaces it is possible to reduce thesolution of the recurrence relation defined by ϕ to r trivial first order recurrence
relations, defining the so-called action variables.

Liouville integrability
A symplectic map satisfying the hypotheses of discrete Liouville theorem is calleda Liouville integrable map.

Liouville integrability implies integrability
Like in the continuous case a Liouville integrable map is a special integrable map
which requires half of the invariants to claim integrability.
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Integrability and growth
In the case of the discrete equations given an initial condition we can always
compute step by step the corresponding sequence.
Then we can intuitively say that integrability is the ability to tell behaviour ofthe equation without the need of computing an infinite sequence.
The concept of integrability is different from the one of solvability.
A very simple example is given by the logistic map:
yn+1 = 4yn (1 − yn ) =⇒ yn =dy1[1 − cos (2n c0 )] =⇒= 2n−1 sin (2n c0 ) .
2
dc0

The discrepancy between to arbitrarily near initial values grows exponentially!
This simple observation led to the conjecture that a good way to establish if
a projective map is integrable or not was to measure how fast the degree withrespect to the initial conditions grows when iterated.
This approach to integrability is called algebraic entropy method [Veselov
1992, Falqui and Viallet 1993 and Bellon and Viallet 1999].
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Definition of algebraic entropy
Algebraic entropy measure the rate of growth of the degrees of the iterates ofa bi-rational map.
Iterating the map ϕ : CPn → CPn we define the sequence of the degrees withrespect to the initial conditions [x0 ]:dk := deg[x0 ] ϕk ,k ∈ N.

The algebraic entropy of this sequence is then defined by the limit:
1
εϕ = lim
log dk .
k→∞ kThis limit always exists due to the properties of bi-rational maps.
Algebraic entropy is a bi-rational invariant of maps.
From the definition if follows that εϕ ≥ 0. Moreover, εϕ = 0 if and only if
dk ' k ν for some ν ∈ N as k → ∞.

Algebraic entropy integrability
We say that a map ϕ : CPn → CPn is integrable according to the algebraic
entropy test if εϕ ≡ 0. Moreover, if dk is periodic (not constant) we say that the
map is periodic, while if dk is linear we say that the map is linearizable.
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Computation of the algebraic entropy
From the definition of algebraic entropy it seems that one has to compute all
the iterates of a map to get its value.
Hopefully enough for most application this is not needed since after a finite
number of iterations the form of the sequence {dk }k∈N stabilizes.
Then to obtain the explicit form of the sequence {dk }k∈N we can then search
for a generating function, i.e. a function such that:g (z) =∞
X

dk z k .

k=0

If we are able to found a generating function then the algebraic entropy isgiven by:ε = log | min {z ∈ C|g (z) has a pole in z} |−1 .
A generating function is a predictive tool which can be used to test thesuccessive members of a finite sequence.

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Duality for bi-rational maps
If the map ϕ possesses k invariants, namely Il with l = 1, . . . , k we can form
the linear combination:
H = α1 I1 + · · · + αl Il .
If degx1 H, degxn H > 1 for an unspecified recurrence[x1 : x2 : · · · : xn : xn+1 ] 7→ [x10 : x1 : . . . , xn−1 : xn+1 ]we have the following factorization:
H ([x0 ]) − H ([x]) = A (x10 , [x])B (x10 , [x]).
Since H is a first integral of ϕ we will have that one of the possible solutionsof H ([x0 ]) − H ([x]) = 0 is just given by [x0 ] = ϕ ([x]).
We can assume that this corresponds to the annihilation of A.
In some circumstances, i.e. when degx1 H, degxn H = 2 we have that also theannihilation of B defines a bi-rational projective map.
We call this map the dual map ϕ∨ [Quispel, Capel and Roberts 2005].
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Duality and first integrals
Assume that the invariants and the map ϕ depends on some arbitrary
constants Il = Il ([x]; a1 , . . . , aM ), for l = 1, . . . , k.
Now assume that choosing some of the ai in a way that there remains N
arbitrary constants and such that for a subset {aij } ⊂ {a1 , . . . , aN } we canwrite H as:
H = ai1 J1 + ai2 J2 + · · · + aiK JK .
Here Ji = Ji ([x]; α1 , . . . , αk ), i = 1, 2, . . . , N are new functions coming fromthe rearrangement of H.
Then using the factorization of H we have that the Ji functions are invariants
for the dual map.

Dual map and invariants
Depending on number of arbitrary constants {aij } the dual map is naturally
equipped with a certain number of invariants.

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Motivations
In a recent paper [Joshi and Viallet, 2017] considered the autonomous limit(2)(2)
of the second member of the dPI and dPII hierarchies, the dPI and dPII
equations.
(2)

(2)

The dPI and dPII equations are given by recurrence relations of orderfour, integrable according to the algebraic entropy approach.
They showed that both maps possesses two first integrals Ilow and Ihigh .
(2)

Using these first integrals they showed that the dual maps of the dPI and(2)
dPII equations are integrable according to the algebraic entropy test andproduced some integrals.
(2)

(2)

Moreover they showed that the integrals of the dPI and dPII equations
share some common properties: in both cases: dp Ilow = (1, 3, 3, 1) anddp Ihigh = (2, 4, 4, 2).
In the same paper they gave a scheme to construct recurrence relations withthe assigned structure of integrals and they produced some new examples outof this construction.
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A classification problem
The problem
We want to make a classification of the fourth-order maps
ϕ : [x : y : z : u : t] 7→ [x 0 : y 0 : z 0 : u 0 : t 0 ] with the following hypotheses:
1

ϕ possesses a polynomial a symmetric first integral Ilow such that
dp Ilow = (1, 3, 3, 1) where the only non-zero coefficients are those appearing(2)(2)
in the Ilow of both dPI and dPII and such that the factorisation0
Ilow ([x ]) − Ilow ([x]) = (x − z)A (x 0 , [x]) holds. This integral has 11 essential
free parameters.

2

ϕ possesses a polynomial symmetric first integral Ihigh such that
dp Ihigh = (2, 4, 4, 2). This integral has 119 essential free parameters.

3

The first integrals Ilow and Ihigh are functionally independent andnon-degenerate.

Symmetric first integral
We say that a first integral ϕ is symmetric if it is invariant under the involution
ι : [x : y : z : u : t] 7→ [u : z : y : x : t].
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Relation to other classifications

The basic idea of this classification is similar to the one of [Quispel, Roberts
and Thompson 1988], but of course in higher dimension.
A possible extension to four dimension of the QRT approach was pursued in[Capel and Sahadevan, 2001], but it this case the author considered
symplectic maps in four dimension with two first integrals of degree pattern(2, 2, 2, 2). Let us notice that first integrals with this degree patterns where
found in the autonomous limit of the fourth order qP equations [Hay, 2007].
In the continuous case the method of imposing an ansatz on the form of thefirst integrals is widely used since the 60s in the theory of superintegrable
systems [Friš, Mandrosov, Smorodinski, Uhlı́ř, and Winternitz, 1965] and yet
produces new and interesting results [Gravel, 2008, Post and Winternitz,
2011].

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A glimpse on how to perform the classification
Find the value of x 0 the relation
Ilow ([x0 ]) − Ilow ([x]) = (x − z)A (x 0 , [x]) ,which is implied by the form of Ilow .
Substitute the obtained form of x 0 into the first integral condition for Ihigh .
We can take coefficients with respect to the independent variables[x : y : z : u : t].
This yield a large system of nonlinear homogeneous equations.
We proceed iteratively to solve the system starting from monomial equations
then reducing the number of equations, until we obtain a collection ofsystems with no new monomial equations.
This leaves us with 117 simpler systems. These systems can be solved to give
25 solutions which, upon relabeling of the parameters and degenerations yield
us 6 fourth order recurrence relations with the desired properties.
We denote by (P.j) for the main maps and by (Q.j) the dual maps where j isa small roman number.
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Properties of the obtained equations
The properties of the obtained equation can be summarized by the following table:
Equation
(P.i)
(Q.i)

dp Ilow(1,3,3,1)(1,2,2,1)

dp Ihigh(2,4,4,2)(2,4,4,2)Degree of growth
cubic
cubic

(P.ii)(Q.ii)(1,3,3,1)–(2,4,4,2)(2,4,4,2)

exponential
exponential(P.iii)(Q.iii)(1,3,3,1)(1,2,2,1)(2,4,4,2)(2,4,4,2)

quadratic
exponential(P.iv) (dPI )(Q.iv)(1,3,3,1)(1,2,2,1), (1,2,2,1)(2,4,4,2)(2,4,4,2)

quadratic
quadratic

(P.v)
(Q.v)

(1,3,3,1)(1,2,2,1), (1,2,2,1)(2,4,4,2)(2,4,4,2)

quadratic
quadratic(1,3,3,1)(1,2,2,1), (1,2,2,1)(2,4,4,2)(2,4,4,2)

quadratic
quadratic

(2)

(2)

(P.vi) (dPII )(Q.vi)
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Selected examples: the map (P.i)
The map [x] 7→ ϕi ([x]) = [x0 ] is given by the following components:
x 0 = −{[νt 2 (x + z) + uz 2 ]y + t 2 µuz + (x + z)2 y 2 }d − at 4 ,y 0 = x 2 d(t 2 µ + xy ),02

u = zxd(t µ + xy ),z 0 = yxd(t 2 µ + xy ),
0

(P.i)

2

t = txd(t µ + xy ).

The map (P.i) has the following degrees of iterates:{dn }P.i = 1, 4, 12, 28, 52, 86, 130, 188, 260, 348, 452, 576, 720, 886, 1074, 1288 . . .

Its generating function is:
gP.i (s) =s 7 − 3s 6 + s 5 − s 4 + 3s 3 + 3s 2 + s + 1.
(s + 1)(s 2 + 1)(s − 1)4

This means that the map (P.i) is integrable according to the algebraic
entropy test with cubic growth. Same considerations apply to the map (Q.i).
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Cubic growth
Besides integrability the cubic growth of (P.i) and (Q.i) gives more
information.
Alongside with information on the structure of the orbit can suggest theexistence of non-algebraic first integrals.
Indeed, if the additional first integral was rational the orbits would be
confined to elliptic curves and the growth of the degree of the iterates would
be quadratic, not cubic [Bellon, 1999].
This means that Ilow and Ihigh can define non-elliptic fibrations as alreadyshown in [Joshi and Viallet, 2017].
Moreover, the maps (P.i) and (Q.i) can be “deflated” to 3-dimensional maps
with two integrals and quadratic growth.
These 3-dimensional maps are Liouville integrable as they possess rank 2
Poisson structures and are reducible to 2-dimensional maps.
The reduction of (Q.i) is a QRT-like map, while the reduction of (P.i) possessa bi-quadratic “anti-invariant” which lies in the framework of [Roberts andJogia, 2015].
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Selected examples: the map (P.ii)
A rather peculiar example is given by the map [x] 7→ ϕii ([x]) = [x0 ]:
x 0 = (x 2 + z 2 )y − uz 2 µ − t 2 (u − 2y ),y 0 = x(t 2 + µx 2 ),022

u = z(t + µx ),z 0 = y (t 2 + µx 2 ),0

2(P.ii)

2t = t(t + µx ).

By construction the map (P.ii) has two integrals and it has the following
degrees of iterates:{dn }P.ii = 1, 3, 9, 21, 45, 93, 189, 381, 765, 1533 . . .

This sequence has generating function:
gP.ii (s) =1 + 2s 2.
(2s − 1)(s − 1)

The main map is non-integrable since it has positive entropy ε = log 2!
Similar considerations hold for the dual map (Q.iii).
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How integrability arises in the list?
We showed that maps from our list can have rather different properties, eventhough all of them possess two invariants.
To understand why some of the maps with two integrals are non-integrable
we need to understand if the discrete Liouville theorem is applicable or not.
It is known from [Byrnes, Haggar, and Quispel, 1999] that from a four
dimensional, measure preserving map with two integrals one can build a a
pre-symplectic structure i.e. a “symplectic structure” where the 2-form ω isdegenerate.
This is not enough to apply the discrete Liouville’s theorem.
So, instead of searching directly for symplectic structures which is very
difficult we searched for discrete Lagrangians for the equations in the list.
Discrete Lagrangians naturally lead to non-degenerate symplectic structures,and vice-versa a symplectic structure can always be inverted to give
Euler-Lagrange equations.
So disproving the existence of a discrete Lagrangian is equivalent todisproving the existence of a symplectic structure.
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Discrete Lagrangians and integrability
A recurrence relation of order 2k in affine coordinates it is said to be
Lagrangian if there exists a function L = L (xn+k , . . . , xn ) such that theEuler-Lagrange equation:
k
X
∂L
l=0

∂xn

(xn+k−l , xn+k−l−1 , . . . , xn−l ) = 0,

is equivalent to the given recurrence relation.
The function L is called the discrete Lagrangian.
We developed a simple and algorithmic test to check if a given equation
possesses or not a discrete Lagrangian.
This study led to understand that the slow growth for equations with no
Lagrangians arises from the existence of other integrals or by the mechanismof deflation.

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Updated properties of the obtained equations
So we can write down an updated table with all the properties of the obtained
solutions:
Equation
(P.i)
(Q.i)

Degree pattern of integrals(1,3,3,1), (2,4,4,2)(1,2,2,1), (2,4,4,2)Degree of growth
cubic (deflatable)
cubic (deflatable)

Lagrangiannono(P.ii)(Q.ii)(1,2,2,1), (1,3,3,1)(2,4,4,2)

exponential
exponential

nono(P.iii)(Q.iii)(1,3,3,1), (2,4,4,2), (2,5,5,2)(1,2,2,1), (2,4,4,2)

quadratic
exponentialnono(P.iv) (dPI )(Q.iv)(1,3,3,1), (2,4,4,2)(1,2,2,1), (1,2,2,1), (2,4,4,2)

quadratic
quadratic

yes
no

(P.v)
(Q.v)

(1,3,3,1), (2,4,4,2)(1,2,2,1), (1,2,2,1), (2,4,4,2)

quadratic
quadratic

yes
no(1,3,3,1), (2,4,4,2)(1,2,2,1), (1,2,2,1), (2,4,4,2)

quadratic
quadratic

yes
no

(2)

(2)

(P.vi) (dPII )(Q.vi)

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SummaryIn this talk we found a certain number of fourth order bi-rational maps
possessing two first integrals of degree pattern (1,3,3,1) and (2,4,4,2).
This family contains integrable, superintegrable and non-integrable members.
These proprieties have been linked and explained through the existence of
discrete Lagrangians.
Search is going on for other fourth order dP equations and their autonomous
limits.
We are also addressing is the geometry of the surfaces defined by theintersections of the first integrals.
Another problem arising regarding these equations is the one ofde-autonomisation. In fact it can be shown that using the “strict” procedureof de-autonomisation several maps do not admit non-autonomous extensions.

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Thank you for your attention!

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