Bi-rational maps in four dimensions with two invariants
aa r X i v : . [ n li n . S I] J un BI-RATIONAL MAPS IN FOUR DIMENSIONS WITH TWOINVARIANTS
GIORGIO GUBBIOTTI, NALINI JOSHI, DINH THI TRAN,AND CLAUDE-MICHEL VIALLET
Abstract.
In this paper we present a class of four-dimensional bi-rationalmaps with two invariants satisfying certain constraints on degrees. We discussthe integrability properties of these maps from the point of view of degreegrowth and Liouville integrability. Introduction
In this paper we classify maps of CP to itself, which possess two polynomialinvariants, under certain conditions. The outcomes of our classification includeeight new classes of maps, which have surprising properties. Despite the existence oftwo invariants, there turn out to be non-integrable cases, with exponential growth.Other cases are integrable, with cubic and quadratic growth. The cases of cubicgrowth are only possible in dimension greater than two [11, 12]. We discuss thegeometric properties of these systems [4].In dimension two it is well-known that integrable bi-rational maps can be charac-terized by the existence of a rational invariant. For instance most of the integrablemaps on the plane fall in the class of QRT maps [33,34], even though there are somenotable exceptions [13,36,45]. The integrability of these maps can be explained ge-ometrically and has led to many interesting developments [13, 37, 41].In higher dimension an analogous general framework does not exist. In partic-ular, for mappings in four dimension, a generalisation of the QRT class [33, 34]was given in [7]. However, this generalisation does not cover all possible integrablemaps in four dimensions. Indeed, some of the new maps obtained in [7] turn out tobe autonomous versions of Painlevé hierarchies [22] which are multiplicative equa-tions in Sakai’s scheme [37]. On the other hand, there exists hierarchies of additive discrete Painlevé equations too [10]. Equations coming from the hierarchies ofadditive Painlevé equations are naturally outside the framework of [7]. Other ex-amples of four-dimensional maps falling outside the class presented in [7] are givenin [8, 9, 21, 25, 30, 31].Our starting point is [25], where the authors considered the autonomous limit of the second member of the d P I and d P II hierarchies [10]. We will denote theseequations as d P (2) I and d P (2) II equations. These d P (2) I and d P (2) II equations are givenby recurrence relations of order four, and shown to be integrable according to thealgebraic entropy approach. Therein the authors showed that both maps possesstwo polynomial invariants. Using these invariants, they produced the dual mapsof the d P (2) I and d P (2) II equations in the sense of [35]. Moreover, they showed that Date : July 2, 2019.2010
Mathematics Subject Classification. these dual maps are integrable according to the algebraic entropy test and alsopossess invariants. In fact, the number of invariants showed that the dual mapsare actually superintegrable . Finally they gave a scheme to construct recurrencerelations of an assigned form. Using this scheme in [25] some new examples, withno classification purposed were presented. Starting from these considerations inthis paper we consider and solve the problem of finding all fourth-order bi-rationalmaps possessing two polynomial invariants of general enough form to contain thoseof the d P (2) I and d P (2) II equations.The structure of the paper is the following: in section 2 we give a concise ex-planation of the background material we need. In particular we discuss the variousdefinitions of integrability for mapping we are going to use throughout the paper.In section 3 we present the motivations for our search and we present our the searchmethod and we state the general result. In section 4 we give the explicit form ofthe maps we derived with the method of section 3 and we discuss their integrabilityproperties following the discussion of section 2. Finally, in section 5 we make somegeneral comments on the maps we obtained, and we underline the possible futuredevelopment. 2. Setting
In this Section we give the fundamental definitions we need to explain how ourlist of equations is found and what kind of integrability we are going to considerwithin this paper. One can also find this setting in our shot communication [21].2.1.
Bi-rational maps and invariants.
The main subject of this paper are bi-rational maps of the complex projective space into itself:(1) ϕ : [ x ] ∈ CP n → [ x ′ ] ∈ CP n , where n > and [ x ] = [ x : x : · · · : x n +1 ] and [ x ′ ] = (cid:2) x ′ : x ′ : · · · : x ′ n +1 (cid:3) to behomogeneous coordinates on CP n . Moreover, we recall that a bi-rational map is arational map ϕ : V → W of algebraic varieties V and W such that there exists arational map ψ : W → V , which is the inverse of ϕ in the dense subset where bothmaps are defined [38].Bi-rational maps of the form (1) are the natural mathematical object needed tostudy autonomous single-valued invertible rational recurrence relations . Indeed, anautonomous recurrence relation of order n is a relation where the ( n + 1) -th elementof a sequence is defined in terms of the preceding n , i.e. an expression of the form:(2) w k + n = f ( w k , . . . , w k + n − ) . A recurrence relation is autonomous if the function f in (2) does not depend explic-itly on k . Moreover, we say that a recurrence relation is rational if the function f in (2) is a rational function. Finally, the recurrence is invertible and single-valued ifequation (2) is solvable uniquely with respect to w . All the terms of the sequence w k for k > n are then obtained by iterated substitution of the previous one intoequation (2). For this reason it is possible to interpret the recurrence relation (2)as a map of the complex space of dimension n into itself as:(3) ̟ : w ∈ C n → w ′ ∈ C n , Bi-rational maps in CP are just Möbius transformations so everything is trivial. I-RATIONAL MAPPINGS IN 4D 3 where w = ( w n , w n − , . . . , w ) are the initial conditions and the map acts as:(4) w ′ = ( f ( w ) , w n , w n − , . . . , w ) . The recurrence relation (2) is then given by the repeated application of the map ̟ ,namely w n + k is the first component of ̟ k . Interpreting the coordinates w ∈ C n as an affine chart in CP n , i.e. assuming that ( w n − , . . . , w ) = [ w n − : · · · : w : 1] we have that the map (3) can be brought to a bi-rational map of CP n into itself ofthe form (1).Throughout the paper we will often make use of the correspondence between bi-rational maps and recurrence relations. This is due to the fact that some definitionsare easier to state and use in the projective setting, while others are easier to stateand use in the affine one. In any case for us “bi-rational map” and “recurrencerelation” will be completely equivalent terms.One characteristic of integrability is the existence of first integrals . In the con-tinuous context, for finite dimensional systems, integrability refers to the existenceof a “sufficiently” high number of first integrals, i.e. of non-trivial functions con-stant along the solution of the differential system. In particular, for a Hamiltoniansystem, the number of first integrals is less as its integrability was given by Li-ouville [27]. In discrete setting, the analogue of first integrals for maps are the invariants which is defined as follows. Definition 1. An invariant of a bi-rational map ϕ : CP n → CP n is a homogeneousfunction I : CP n → C such that it is preserved under the action of the map, i.e.(5) ϕ ∗ ( I ) = I, where ϕ ∗ ( I ) means the pullback of I through the map ϕ , i.e. ϕ ∗ ( I ) = I ( ϕ ([ x ])) .For n > , an invariant is said to be non-degenerate if:(6) ∂I∂x ∂I∂x n = 0 . Otherwise an invariant is said to be degenerate .In what follows we will concentrate on a particular class of invariants:
Definition 2.
An invariant I is said to be polynomial , if in the affine chart [ x : · · · : x n : 1] the function I is a polynomial function.In definition 2 we use x n +1 as homogenising variable to go from an affine (polyno-mial) form to a projective (rational) form of the invariants. A polynomial invariantin the sense of definition 2 written in homogeneous variables is always a ratio-nal function homogeneous of degree 0. The form of the polynomial invariant inhomogeneous coordinates is then given by:(7) I ([ x ]) = I ′ ([ x ]) x dn +1 , d = deg I ′ ([ x ]) , where deg is the total degree.To better characterize the properties of these invariants we introduce the follow-ing: Definition 3.
Given a polynomial function F : CP n → V , where V can be either CP n or C , we define the degree pattern of F to be:(8) dp F = (cid:0) deg x F, deg x F, . . . , deg x n F (cid:1) . G. GUBBIOTTI, N. JOSHI, D. T. TRAN, AND C-M. VIALLET
Finally we will consider invariants which are not of generic shape, but satisfythe following condition:
Definition 4.
We say that a function I : CP n → C is symmetric if it is invariantunder the following involution:(9) ι : [ x : x : · · · : x n : x n +1 ] → [ x n : x n − : · · · : x : x n +1 ] , i.e. ι ∗ ( I ) = I .2.2. Integrability of bi-rational maps.
Integrability both for continuous anddiscrete systems can be defined in different ways, see [23, 46] for a complete discus-sion of the continuous and the discrete case. Different ways of defining integrabilitydo not always necessarily agree, even though most of the time they do. We under-line that the list we are going to make is not meant to be completely exhaustiveof all the possible definitions of integrability. We will discuss only the definitionsfor autonomous recurrence relations we will need throughout the rest of the pa-per. We mention that additional definitions of integrability have been proposed fornon-autonomous systems.In general the solution of a recurrence relation of order n will depend on n arbitrary constants. This means that if a recurrence relation defined by the map ϕ : CP n → CP n possesses n − invariants I j , j = 1 , . . . , n − , then, in principle, itis possible to reduce it to a map ˆ ϕ : CP → CP by solving the relations:(10) I j = κ j , where κ j are the value of the invariants on a set of initial data. This stimulates thesimplest and most natural definition of integrability for maps: Definition 5 (Existence of invariants) . An n -dimensional map is (super)integrableif it admits n − functionally independent invariants . Remark . We underline that, in general, the reduction to a lower-dimensional mapsolving the system of equations (10) can break the bi-rationality.Definition 5 is very general, and works for arbitrary maps. If some additionalstructure are present, then the number of invariants needed for integrability can besignificantly reduced. A special, but relevant case is the one of Poisson maps.
Definition 6 (Poisson structures and Poisson maps [7, 29]) . In affine coordinates w a Poisson structure of rank r is a skew-symmetric matrix J = J ( w ) of constantrank r such that the Jacobi identity holds :(11) n X l =1 (cid:18) J li ∂J jk ∂w l − + J lj ∂J ki ∂w l − + J lk ∂J ij ∂w l − (cid:19) = 0 , ∀ i, j, k. A Poisson structure defines a
Poisson bracket through the identity:(12) { f, g } = ∇ f J ( w ) ∇ g T , where ∇ f is the gradient of f . Two functions f and g are said to be in involution with respect to the Poisson structure J ( w ) if { f, g } = 0 . We can easily see that { w i − , w j − } = J ij . A map of the affine coordinates ϕ : w w ′ is a Poisson map if it preserves the Poisson structure J ( w ) , i.e. if:(13) d ϕJ ( w ) d ϕ T = J ( w ′ ) , I-RATIONAL MAPPINGS IN 4D 5 where d ϕ is the Jacobian matrix of the map ϕ .Then we have the following characterisation of integrability for Poisson maps: Definition 7 (Liouville integrability [5, 28, 42]) . An n -dimensional Poisson mapis integrable if it possesses n − r functionally independent invariants in involutionwith respect to this Poisson structure . Remark . A Poisson structure of full rank, i.e. n = 2 r is invertible. The inversematrix of the matrix J ( w ) , i.e. Ω ( w ) = J − ( w ) is said to be a symplectic structure .We note that in the symplectic case we only need n/ invariants in involution toclaim integrability.Symplectic structures are quite important in the theory of integrable maps. Forinstance, the classification made in [7] was carried out assuming of the existence of linear Poisson structure and of two invariants.A difficult problem is, given a map, to find if there exists a symplectic struc-ture for which this map is symplectic. In [6] it was proved that there exists a pre-symplectic structure (a degenerate sympectic structure) for any n -dimensionalvolume-preserving map possessing n − invariants. The rank of the obtained pre-symplectic structure is n − which implies that to claim integrability in the sense ofLiouville one must be able to find another invariant. On the other hand, when themap comes from a discrete variational principle , i.e. it is variational , to find a sym-plectic structure is easy. We recall that an even-order recurrence relation (2) is saidto be variational if there exists a function, called Lagrangian , L = L ( w k + N , . . . , w n ) such that the recurrence relation (2) is equivalent to the Euler-Lagrange equations:(14) N X i =0 ∂L∂w k ( w k + N − i , . . . , w k − i ) = 0 . Here N = n/ in the recurrence (2). A Lagrangian is called normal if(15) ∂ L∂w k ∂w k + N = 0 . Let T be a shift operator, i.e T j ( w k + i ) = w k + i + j . Then due to the normalitycondition the discrete Ostrogradsky transformation [40]:(16) O : w → ( q , p ) where the new coordinates ( q , p ) = ( q , . . . , q N , p , . . . , p N ) are defined through theformula: q i = w k + i − , i = 1 , . . . , N, (17a) p i = T − N − i X j =0 T − j ∂L∂w j + i , i = 1 , . . . , N, (17b)is well defined and invertible. Then the following result holds true [5]: Lemma 1.
The map given by
Φ =
O ◦ ̟ ◦ O − : ( q , p ) → ( q ′ , p ′ ) , where ̟ is themap corresponding to the Euler-Lagrange equations (14) has the following form q ′ i = q i +1 , i = 1 , , . . . , N − , (18a) q ′ N = α ( q , p ) , (18b) G. GUBBIOTTI, N. JOSHI, D. T. TRAN, AND C-M. VIALLET p ′ i = p i +1 + ∂ e L∂q i +1 ( q , p ) , i = 1 , , . . . , N − , (18c) p ′ N = ∂ e L∂τ ( q, p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) τ = α ( q ,p ) , (18d) where α ( q , p ) is the solution with respect to q N ′ of the equation: (19) p = − ∂L∂q ( q , q ′ N ) , and e L ( q , p ) = L ( q , α ( q , p )) . Moreover, the map (18) is symplectic with respect tothe canonical symplectic structure: (20) Ω = (cid:18) O N I N − I N O N (cid:19) , where O N is the zero N × N matrix and I N the N × N identity matrix. Lemma 1 has the following corollary:
Corollary 2.
The Euler–Lagrange equations (14) admit the following non-degeneratePoisson bracket: (21) J ( w ) = d O − Ω − (d O − ) T , where the differential of the Ostrogradsky transformation O must be evaluated onthe original coordinates. Therefore we have that corollary 2 allows us to construct a non-degenerate Pois-son structure, and hence a symplectic structure, for every variational map .Lagrangians for N -order recurrence relations can be found following [24] or [20]for N > . The method presented in [20] allows also to disprove the existence of aLagragian for a given N -order recurrence relation for N > .Moreover, bi-rational maps possess another definition of integrability: the lowgrowth condition [3, 15, 43]. To be more precise we say that an n -dimensional bi-rational map is integrable if the degree of growth of the iterated map ϕ k is poly-nomial with respect to the initial conditions [ x ] . Therefore we have the followingcharacterisation of integrability: Definition 8 (Algebraic entropy [3]) . An n -dimensional bi-rational map is inte-grable in the sense of the algebraic entropy if the following limit(22) ε = lim k →∞ k log deg [ x ] ϕ k , called the algebraic entropy is zero for every initial condition [ x ] ∈ CP n .Algebraic entropy is an invariant of bi-rational maps, meaning that its value isunchanged up to bi-rational equivalence. Practically algebraic entropy is a mea-sure of the complexity of a map, analogous to the one introduced by Arnol’d [2]for diffeomorphisms. In this sense growth is given by computing the number ofintersections of the successive images of a straight line with a generic hyperplanein complex projective space [43].The value of the degree of the iterates of the map is conditioned by its singularitystructure . Some hypersurfaces are blown down by the map. If one of the successiveimages of these hypersurfaces coincide with a singular variety, there is a drop in I-RATIONAL MAPPINGS IN 4D 7 the degree [3, 39, 44]. Therefore, from a heuristic point of view we can say that thesingularity makes the entropy. This actually also applies to non-autonomous caseslike the discrete Painlevé equations [37].In principle, the definition of algebraic entropy in equation (22) requires us tocompute all the iterates of a bi-rational map ϕ to obtain the sequence(23) d k = deg [ x ] ϕ k , k ∈ N . Fortunately, for the majority of applications the form of the sequence can be inferredby using generating functions [26]:(24) g ( z ) = ∞ X n =0 d k z k . A generating function is a predictive tool which can be used to test the successivemembers of a finite sequence. It follows that the algebraic entropy is given by thelogarithm of the smallest pole of the generating function, see [18, 19].Several results are known about the relationship of the above definitions of in-tegrability. First of all, the low growth condition means that the complexity of themap is very low, and it is known that invariants help in reducing the complexity ofa map. Indeed the growth of a map possessing invariants cannot be generic sincethe motion is constrained to take place on the intersection of hypersurfaces definedby the invariants. However, the drop in complexity must be big enough to reducethe growth to a polynomial one. On the other hand it is known that the existenceof invariants can give some bounds on the growth of bi-rational maps. Indeed, it isknown that the orbits of superintegrable maps with rational invariant are confinedto elliptic curves and the growth is at most quadratic [4,17]. In low dimension someexplicit results on the growth of bi-rational maps are known. For maps in CP , itwas proved in [12] that the growth can be only bounded, linear, quadratic or expo-nential. Linear cases are trivially integrable in the sense of invariants. We note thatfor polynomial maps in C , it was already known from [43] that the growth can beonly linear or exponential. It is known that QRT mappings and other maps withinvariants in CP possess quadratic growth [13], so the two notions are actuallyequivalent for large class of integrable systems.2.3. Duality.
Now we discuss briefly the concept of duality for rational maps,which was introduced in [35]. Let us assume that our map ϕ possesses L invariants,i.e. I j for j ∈ { , . . . , L } . Then we can form the linear combination:(25) H = α I + · · · + α L I L . Being a function of invariants it follows that H defined by (25) is itself an invariantof the map. Remark . We note that in principle more general combinations of invariants canbe considered:(26) H = P d ( I , I , . . . , I L ) where P d is a homogeneous polynomial of total degree d in L variables. Again evenin this generalized case H defined by (26) is an invariant of the map. However, inthis paper we won’t consider this case, following the original definition of [35]. G. GUBBIOTTI, N. JOSHI, D. T. TRAN, AND C-M. VIALLET
For an unspecified recurrence relation(27) [ x : x : · · · : x n +1 ] (cid:2) x ′ : x ′ : · · · : x ′ n +1 (cid:3) = [ x ′ : x : · · · : x n +1 ] we can write down the invariant condition for H (25):(28) b H ( x ′ , [ x ]) = H ([ x ′ ]) − H ([ x ]) = 0 . Since we know that [ x ′ ] = ϕ ([ x ]) is a solution of (28) we have the following factor-ization:(29) b H ( x ′ , [ x ]) = A ( x ′ , [ x ]) B ( x ′ , [ x ]) . We can assume without loss of generality that the map ϕ corresponds to the an-nihilation of A in (29). Now since deg x ′ b H = deg x H and deg x n b H = deg x n H we have that if if deg x H, deg x n H > the factor B in (29) is non constant . Ingeneral, since the map ϕ is bi-rational, we have the following equalities: deg B x ′ = deg x ′ b H − deg x ′ A = deg x H − , (30a) deg B x n = deg x n b H − deg x n A = deg x n H − . (30b)Therefore we have that, in general, if deg x H, deg x n H > , the annihilationof B does not define a bi-rational map, but an algebraic one. However when deg x H, deg x n H = 2 the annihilation of B defines a bi-rational projective map.We call this map the dual map and we denote it by ϕ ∨ . Remark . We note that in principle for deg x H = deg x n H = d > , more generalfactorizations can be considered:(31) b H ( x ′ , [ x ]) = d Y i =1 A i ( x ′ , [ x ]) , but in this paper we won’t consider this case.Now assume that the invariants (and hence the map ϕ ) depends on some arbitraryconstants I i = I i ([ x ]; a i ) , for i = 1 , . . . , K . Choosing some of the a i in such a waythat there remains M arbitrary constants and such that for a subset a i k we canwrite equation (25) in the following way:(32) H = a i J + a i J + · · · + a i K J a iK , where J i = J i ([ x ]) , i = 1 , , . . . , K are new functions. Then using the factorization(29) we have that the J i functions are invariants for the dual maps. Remark . It is clear from equation (32) that even though the dual map is naturallyequipped with some invariants, it is not necessarily equipped with a sufficient num-ber of invariants to claim integrability. In fact there exists examples of dual mapswith any possible behaviour, integrable, superintegrable and non-integrable [21,25]. We remark that this assertion is possible because we are assuming that all the invariants arenon-degenerate. It is easy to see that degenerate invariants can violate this property.
I-RATIONAL MAPPINGS IN 4D 9 Derivation of the class of 4D maps
In this Section we explain how we derive the class of 4D maps with two invariantswe are going to present in Section 4.Our starting points are the maps corresponding to the autonomous d P I and the d P II equations and their invariants as presented in [25]. These two are maps of CP into itself with coordinates [ x : y : z : u : t ] . Their components are given by:( d P I ) x ′ = − ay (cid:0) x + y + z + 2 yz + 2 xy + xz + zu (cid:1) − bty ( y + z + x ) − cyt + dt ,y ′ = ayx , z ′ = axy , u ′ = axyz, t ′ = axyt. and by:( d P II ) x ′ = dt − a ( t − y ) ( t + y ) (cid:0) ut − yz − uz − yxz − x y (cid:1) − cyt − bt ( t − y ) ( t + y ) ( z + x ) ,y ′ = ax (cid:0) t − y (cid:1) (cid:0) t − x (cid:1) , z ′ = ay (cid:0) t − y (cid:1) (cid:0) t − x (cid:1) ,u ′ = az (cid:0) t − y (cid:1) (cid:0) t − x (cid:1) , t ′ = at (cid:0) t − y (cid:1) (cid:0) t − x (cid:1) . It can be checked that the map d P I has two invariants I (I) and I (I) which are: t I (I) = ayz (cid:0) − y − yz − xy − z − zu + xu (cid:1) − btyz ( z + y ) − cyzt + dt ( z + y ) , (33a) t I ( I ) = ayz (cid:0) zu + xy + y + 2 yz + z (cid:1) ( z + u + y + x )+ cyz ( z + u + y + x ) t − d (cid:0) zu + xy + y + 2 yz + z (cid:1) t + byz ( y + z + x ) ( u + y + z ) t, (33b)while the map d P II possesses two invariants I (II) and I (II) given by t I (II) = a ( t − z ) ( t + z ) ( t − y ) ( t + y ) ( ux − uz − xy − yz ) − bt (cid:0) z t + t y − z y (cid:1) − ct yz + dt ( z + y ) , (34a) t I (II) = a (cid:2)(cid:0) u + z + y + x (cid:1) t − z y ( uz + xy + yz ) − (2 yuz + 2 uzxy + x z + 2 xzy + 2 x y + u y + 2 z y + 2 u z ) t + (2 x y z + 2 uy z + 2 xy z + 2 yuz + z y + y z + 2 u y z + x y + 2 uxy z + 2 uxyz + 2 xzy + z u ) t (cid:3) + bt ( t − z ) ( t + z ) ( t − y ) ( t + y ) ( z + x ) ( u + y )+ ct (cid:0) xzt − z y + yut − yuz − xzy (cid:1) − dt (cid:0) xt + zt − zy − xy − uz + ut − yz + yt (cid:1) . (34b)The invariants of the maps d P I and d P II have the following properties: Property A:
The invariants are symmetric in the sense of definition 4.
Property B:
The lowest order invariants (33a) and (34a) have degree pattern (1 , , , and are particular instances of the homogeneous poly-nomial in C [ x, y, z, u, t ] :(35) t I low = t ( y + z ) s − t ( ux − uz − xy ) s + s t yz + t ( y + z ) s + t yz ( y + z ) s + t ( y + z )( ux − uz − xy ) s − t yz ( ux − uz − xy ) s + s t y z + t yz ( y + z ) s − y z ( ux − uz − xy ) s + s y z , depending parametrically on 11 coefficients, namely s i , i = 1 , . . . , . Property C:
The highest order invariants (33b) and (34b) have degree pat-tern (2 , , , . The most general homogeneous polynomial in C [ x, y, z, u, t ] depends parametrically on 1820 coefficients. Tak-ing into account the symmetry with respect to the involution (9)the number of coefficients 121. Since, one of this coefficients isjust an additional constant then we can lower the number of in-dependent coefficient to 120. We denote this invariant by I high ,but we do not present the general form of this polynomial here,since it will be too cumbersome to write down.Based on the above consideration it is natural to address the following problem: Problem 1.
Find all the bi-rational maps ϕ : CP → CP and their dual maps ϕ ∨ : CP → CP having two non-degenerate, functionally independent invariantswith properties A, B and C.Solving this problem amount to obtain a list of equations which are expected tobehave like the two fourth-order Painlevé equations d P I and d P II . Before going tothe solution of this problem, let us remark the following general result on the dualmap of a map with two invariants possessing properties A, B and C: Lemma 3.
Assume that a map ϕ : CP → CP possesses two invariants withproperties A, B and C. Then we have the map ϕ has degree pattern dp ϕ = (2 , , , and the maximal degree pattern of the dual ϕ ∨ : CP → CP is dp ϕ ∨ = (2 , , , .Proof. By direct computation is it possible to check that if an invariant I low hasthe form (35) then the invariant condition (5) implies the following factorisation:(36) I low ([ x ′ ]) − I low ([ x ]) = ( x − z ) A ( x ′ , [ x ]) . Equation (36) means that we have the following degree distribution:(37) deg x ′ deg x deg y deg z deg u ϕ ∗ ( I low ) 1 3 3 1 0 I low A The second part of the statement comes from an analogous consideration appliedto equation (29). Since the degree pattern of A is fixed, the degree pattern of B ismaximal when there are no factors depending only on [ x ] . Under this assumptions I-RATIONAL MAPPINGS IN 4D 11 we find the following distribution of the degrees:(38) deg x ′ deg x deg y deg z deg u ϕ ∗ ( H ) 2 4 4 2 0 H A B This ends the proof. (cid:3)
Corollary 4.
Bi-rational maps possessing two invariants satisfying properties A,B, and C in general are not self-dual . We sketch now the procedure we used to solve problem 1. We underline thatthis procedure is based on the one proposed in [25] to find bi-rational maps withinvariants of assigned degree pattern.(1) Find the value of x ′ from (36) where I low is given by equation (35).(2) Substitute the obtained form of x ′ into the invariant condition (5) for I high .Geometrically this describes the intersection of the two hypersurfaces givenby I low = I (0) low and I high = I (0) hight , where I (0) low and I (0) hight are arbitrary con-stants.(3) We can take coefficients with respect to the independent variables. Thisyield a system of nonlinear homogeneous equations. We put this system ina collection of systems that we call.(4) We convert this system to a set of simpler systems by solving iterativelyall the monomial equations of each system. At each stage we exclude thesystems originating invariants contraddicting properties A, B and C.(5) This yields 117 different smaller systems.(6) Solving these systems we found 25 solutions respecting the properties A, Band C.Through a degeneration scheme the 25 solutions we obtain can be cast into sixdifferent maps along with their duals. We proved the following: Theorem 5.
The solutions of problem 1, up to degeneration and identification ofthe free parameters, is given by six pairs of main/dual maps which we denote by(P.x) with x small roman number for the main maps and by (Q.x) for the dual map.
We call this class of maps the (P,Q) class . In the next section we present theexplicit form of these maps and we discuss their integrability properties.4.
Maps of the (P,Q) class and their integrability properties
In this section we show the explicit form of the maps of the class (P,Q). We denotethe pairs of main/dual maps by (x) where x is a small roman number. Moreoverwe discuss their integrability properties from the point of view of the existence ofinvariants, the degree growth of their iterates and the existence of Lagrangians. Forthe cases admitting Lagrangian following corollary 2 we present the form of theirsymplectic structure.
Maps (i).
The main map [ x ] ϕ i ([ x ]) = [ x ′ ] has the following components:(P.i) x ′ = −{ [ νt ( x + z ) + uz ] y + t µuz + ( x + z ) y } d − at ,y ′ = x d ( t µ + xy ) , z ′ = yxd ( t µ + xy ) ,u ′ = zxd ( t µ + xy ) , t ′ = txd ( t µ + xy ) . This map depends on four parameters a, d and µ, ν . The map (P.i) has the followingdegrees of iterates:(39) { d n } P.i = 1 , , , , , , , , , , , , , , , , , , . . . with generating function:(40) g P.i ( s ) = s − s + s − s + 3 s + 3 s + s + 1( s + 1)( s + 1)( s − . All the poles of the generating function (40) lie on the unit circle, so that the map(P.i) is integrable according to the algebraic entropy criterion. Moreover, due tothe presence of ( s − in the denominator of the generating function (40) we havethat the main map (P.i) has cubic growth.The map (P.i) has the following invariants: t I P.ilow = at yz + d (cid:2) νy z − yz ( ux − uz − xy ) µ (cid:3) t − y z d ( ux − xy − yz − uz ) , (41a) t I P.ihigh = [( uz + xy − yz ) µ − νyz ] at + (cid:2) yz ( xy + yz + uz ) a + d ( uz + xy − yz ) µ +2 dyz ( ux − yz ) µν − dν y z (cid:3) t + (cid:2) dzy ( uz + xy − yz )( xy + yz + uz ) µ + 2 dy z νux (cid:3) t + y z d ( xy + yz + uz ) . (41b)The two invariants (41) alone cannot explain why the map (P.i) is integrableaccording to the algebraic entropy criterion. Indeed, as we stressed in section 2 twoinvariants are not enough to claim Liouville integrability, nor to claim integrabilityin the sense of definition 5. We can show using the method presented in [20] thatthe map (P.i) is not variational. However, as we shown in [21], the recurrenceassociated to the map (P.i) can be deflated to a three dimensional map via thetransformation v k = w k w k +1 . The map obtained in this way is then integrable inthe sense of Liouville. For all the details we refer to [21].The dual map [ x ] ϕ ∨ i ([ x ]) = [ x ′ ] has the following components:(Q.i) x ′ = [ β (2 xy − yz + uz ) µ + ( βν − α ) y ( x − z )] t + βy ( z y − x y + uz ) y ′ = x β ( t µ + xy ) , z ′ = yxβ ( t µ + xy ) ,u ′ = zxβ ( t µ + xy ) , t ′ = txβ ( t µ + xy ) . This map depends on three parameters α, β , and µ, ν . The parameters µ and ν are shared with the main map (P.i). The map (Q.i) has the following degrees of I-RATIONAL MAPPINGS IN 4D 13 iterates:(42) { d n } Q.i = 1 , , , , , , , , , , , , , . . . with generating function:(43) g Q.i ( s ) = ( s − s − s − s − s − s + s + 1)( s − . This means that the dual map is integrable according to the algebraic entropy testwith cubic growth, just like the main map.The main map (P.i) possesses two invariants and depends on a and d whereasthe dual map (Q.i) do not depend on them. Then according to (32) we can writedown the first invariants for the dual map (Q.i) as:(44) αI P.ilow + βI P.ihigh = aI Q.ilow + dI Q.ihigh . Therefore we obtain the following expressions: t I Q.ilow = ( yzα + ( µxy − yzµ − yνz + µuz ) β ) t + βyz ( xy + yz + uz ) , (45a) t I Q.ihigh = (cid:8)(cid:2) y z ν − yz ( ux − uz − xy ) µ (cid:3) α + (cid:2) ( uz + xy − yz ) µ + 2 yz ( ux − yz ) νµ − ν y z (cid:3) β (cid:9) t + (cid:8) z y ( xy + yz − ux + uz ) α + (cid:2) yz ( uz + xy − yz )( xy + yz + uz ) µ + 2 y z νux (cid:3) β (cid:9) t + z y ( xy + yz + uz ) β. (45b)The invariant (45a) has degree pattern (1 , , , .The properties of the dual map (Q.i) are very similar to those of the main map(P.i). Again, the two invariants (45) alone cannot explain the low growth and fol-lowing [20] no Lagrangian exists. However, as was shown in [21], the recurrenceassociated to the map (Q.i) can be deflated to a three dimensional via the trans-formation v k = w k w k +1 . Again, the map obtained in this way is integrable in thesense of Liouville. For all the details we refer again to [21].4.2. Maps (ii).
The main map [ x ] ϕ ii ([ x ]) = [ x ′ ] has the following components:(P.ii) x ′ = (cid:2) ( x + z ) y − uz (cid:3) µ − t ( u − y ) ,y ′ = x ( t + µx ) , z ′ = y ( t + µx ) ,u ′ = z ( t + µx ) , t ′ = t ( t + µx ) . This map depends on the parameter µ . The map (P.ii) has the following degreesof iterates:(46) { d n } P.ii = 1 , , , , , , , , , . . . with generating function:(47) g P.ii ( s ) = 1 + 2 s (2 s − s − . This means that the main map is non-integrable according to the algebraic entropytest with positive entropy ε = log 2 . Despite being non-integrable the main map (P.ii) has, by construction, the fol-lowing invariants: t I P.iilow = ( x − z ) ( u − y ) (cid:0) t + z µ (cid:1) (cid:0) µy + t (cid:1) (48a) t I P.iihigh = h ( x − z ) y + y z − yz u + u z i µ + 2 t (cid:2)(cid:0) x − xz + 2 z (cid:1) y − yz u + u z (cid:3) µ + t (cid:0) z + u + x + y − uy − xz (cid:1) (48b)Moreover, using the test of [20] we have that the map (P.ii) is not variational.The dual map [ x ] ϕ ∨ ii ([ x ]) = [ x ′ ] is given by the following components:(Q.ii) x ′ = α (cid:2)(cid:0) x − z (cid:1) y + uz (cid:3) µ + t αu + βy ( x − z ) µ + t β ( x − z ) ,y ′ = αx (cid:0) t + µx (cid:1) , z ′ = αy (cid:0) t + µx (cid:1) ,u ′ = αz (cid:0) t + µx (cid:1) , t ′ = αt (cid:0) t + µx (cid:1) . This map depends on three parameters α, β and µ . The parameter µ is shared withthe main map (P.ii). The map given by (Q.ii) has the following degrees of iterates:(49) { d n } Q.ii = 1 , , , , , , , , , , . . . with generating function:(50) g Q.ii ( s ) = 1 + 2 s (2 s − s − . This means that the main map is non-integrable according to the algebraic entropytest with positive entropy ε = log 2 . We remark that the growth is exactly the sameas the main map.Since the main map (P.ii) possesses two invariants, but it has only one parameter µ shared with the dual map. Then according to (32) we can only write down asingle invariant for the dual map (Q.ii) as:(51) I Q.ii = αI P.iihigh + βI P.iilow . The invariant (51) has degree pattern (2 , , , .Finally, using the test of [20] we have that the map (Q.ii) is not variational, asthe main map (P.ii).For additional comments about the maps (P.ii) and (Q.ii) we refer to [21].4.3. Maps (iii).
The main map [ x ] ϕ iii ([ x ]) = [ x ′ ] is has the following compo-nents:(P.iii) x ′ = − at − µd ( z + x + y ) t + νd [2( x + z ) y + uz ] t + d (2 yz u + 2 y zx ) ,y ′ = x d ( νt + 2 xy ) , z ′ = yxd ( νt + 2 xy ) ,u ′ = zxd ( νt + 2 xy ) , t ′ = txd ( νt + 2 xy ) . This map depends on four parameters a, d and µ, ν . The map (P.iii) has the fol-lowing degrees of iterates:(52) { d n } P.iii = 1 , , , , , , , , , , , , , , , , , , , , . . . I-RATIONAL MAPPINGS IN 4D 15 with generating function:(53) g P.iii ( s ) = − s + 4 s + 10 s + 9 s + 13 s + 7 s + 6 s + 2 s + 1( s − s + 1)( s + s + 1) ( s − . This means that the main map is integrable according to the algebraic entropy testwith quadratic growth.The main map (P.iii) has the following invariants: t I P.iiilow = 2 at yz + 2 yzµd ( y + z ) t − yzdν ( − yz − xy − uz + ux ) t − y z ( ux − uz − xy ) d, (54a) t I P.iiihigh = 4 µa ( y + z ) t + (4 dyzµ − ayzν + 4 dz µ + 4 dy µ + 2 azνu + 2 ayνx ) t + 2 µνd (2 uz + 2 xy + yzx + zuy ) t + ( dν x y − dν y z + 4 dν uxyz + dν u z + 4 ay zx + 4 ayz u ) t + 4 µdyz (2 uz + 2 xy + yzx + zuy ) t + 2 ydνz ( uz + 2 xy )(2 uz + xy ) t + 4 y z d ( u z + x y + uxyz ) . (54b)Moreover, we note that according to the test in [20] the main map (P.iii) doesnot posses a Lagrangian. However, by direct search, we can prove that this maphas an additional functionally independent invariant of degree pattern (2 , , , .This means that the low growth of the main map (P.iii) is explained in terms ofintegrability as existence of invariants, as given by definition 5. More specifically,the quadratic growth it explained by the fact that if a map in CP has threerational invariants, the orbits are confined to elliptic curves and the growth is atmost quadratic [4].The dual map [ x ] ϕ ∨ iii ([ x ]) = [ x ′ ] has the following components:(Q.iii) x ′ = 2 µβ ( z − x ) t + { βν [ zu + 2 y ( x − z )] + 2 αy ( z − x ) } t + 2 βz yuy ′ = x β ( νt + 2 xy ) , z ′ = yxβ ( νt + 2 xy ) ,u ′ = zxβ ( νt + 2 xy ) , t ′ = txβ ( νt + 2 xy ) . This map depends on four parameters α, β and µ.ν . The parameters µ, ν are sharedwith the main map (P.iii). The map (Q.iii) has the following degrees of iterates:(55) { d n } Q.iii = 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , . . ., with generating function:(56) g Q.iii ( s ) = P Q.iii ( s ) Q Q.iii ( s ) , where P Q.iii ( s ) = s + 2 s + 4 s + 6 s − s + 5 s + s + 5 s + s + 4 s + s + 3 s + s + 1 (57a) Q Q.iii ( s ) = (1 − s ) (cid:0) s + 1 (cid:1) (cid:0) s − s − s + 1 (cid:1) . (57b)The growth of the map (Q.iii) is given by the inverse of the smallest pole ofthe generating function (56). These poles are given by the zeroes of the function Q Q.iii ( s ) in (57b). Clearly the zeroes of − s and s + 1 , lie on the unit circle,therefore we have to look at the location of the zeroes of the polynomial:(58) q ( s ) = s − s − s + 1 . Defining q ( s ) = − s + 1 and q ( s ) = s − s we have that on the circle C ρ := { s ∈ C || s | = ρ } with ρ ∈ (1 / , the following inequality holds:(59) | q ( s ) | < | q ( s ) | . By Rouche’s theorem [1] this implies that q ( s ) and q ( s ) + q ( s ) = q ( s ) havethe same number of zeroes inside the circle C ρ , i.e. the polynomial Q Q.iii ( s ) hasa unique zero inside the circle C ρ . This zero is the smallest one of Q Q.iii ( s ) anddue to the fact that Q Q.iii ( s ) has real coefficients this zero is real. This implies thegrowth of the dual map (Q.iii) is exponential . The approximate value of the zero of Q Q.iii ( s ) inside C ρ is s = 0 . . . . . This implies that the algebraicentropy of the dual map (Q.iii) is:(60) ε Q.iii = log (2 . . . . ) . The growth of the sequence of degrees of equation (Q.iii) is then slightly greaterthan n .Since the main map (P.iii) possesses two invariants and depends on a and d whereas the dual map (Q.iii) do not depend on them according to (32) we canwrite down the invariants for the dual map (Q.iii) as:(61) αI P.iiilow + βI P.iiihigh = aI Q.iiilow + dI Q.iiihigh . Therefore we obtain the following expressions: t I Q.iiilow = 2 µ ( y + z ) βt + [2 αyz + ν ( uz + xy − yz ) β ] t + 2 yz ( uz + xy ) β, (62a) t I Q.iiihigh = 2 µ ( yz + z + y ) βt + (cid:2) yµz ( y + z ) α + µν (2 uz + 2 xy + yzx + zuy ) β (cid:3) t + 12 [ νyz (2 yz + xy + uz − ux ) α + ν (4 uxyz + u z − z y + x y ) β ] t + 2 µyz (2 uz + 2 xy + yzx + zuy ) βt + ( z y ( uz + xy − ux ) α + νyz ( uz + 2 xy )(2 uz + xy ) β ) t + 2 y z ( u z + x y + uxyz ) β. (62b)The first invariant (62a) has degree pattern (1 , , , and the second invariant hasdegree pattern (2 , , , . However, the degree pattern of the second invariant isnot minimal : we can reduce the degree pattern of the second invariant to (1 , , , I-RATIONAL MAPPINGS IN 4D 17 by replacing I high with βI high − I low . Moreover, we can see that the existenceof these two invariants is not sufficient to ensure the low growth of the dual map(Q.iii). Finally, we note that that the dual map (Q.iii) according to the test in [20]does not possess a Lagrangian.4.4. Maps (iv).
The main map [ x ] ϕ iv ([ x ]) = [ x ′ ] has the following compo-nents:(P.iv) x ′ = − t a − bt y − dνy ( x + y + z ) t − dy ( y + 2 xy + 2 yz + x + xz + uz + z ) ,y ′ = dyx , z ′ = dxy , u ′ = dzxy, t ′ = dtxy. This map depends on four parameters a, b, d and ν . We note that the map (P.iv)is the autonomous d P I , derived in [10] and whose invariants, duality and growthproperties where studied in [25]. For sake of completeness we repeat these propertieshere. The map (P.iv) has the following degrees of iterates:(63) { d n } P.iv = 1 , , , , , , , , , , , , , , , . . . with generating function:(64) g P.iv ( s ) = − s − s − s + 2 s + 2 s + s + 1( s + 1)( s − . This means that the main map is integrable according to the algebraic entropy testwith quadratic growth.The map (P.iv) has the following invariants: t I P.ivlow = t ( y + z ) a + zbt y + dνyz ( y + z ) t − dyz ( ux − xy − yz − uz − y − z ) , (65a) t I P.ivhigh = − νa ( y + z ) t + (cid:2)(cid:0) y + z + 2 yz + uz + xy (cid:1) a − yzbν (cid:3) t − yz (cid:2) ν d ( y + z ) − b ( y + z + u + x ) (cid:3) t + dνyz ( uy + xz + 2 ux ) t + dzy ( x + u + z + y )( y + z + 2 yz + uz + xy ) . (65b)Using the methods of [20] we have that the map (P.iv) is variational. In affinecoordinates w n its Lagrangian is given by:(66) L P.iv = w n w n +1 w n +2 + w n w n +1 w n + w n +1 w n + ν (cid:18) w n w n +1 w n (cid:19) + ad log ( w n ) + bd w n . Using Corollary 2, we obtain the following non-degenerate Poisson bracket (67) J P.iv = dw n − − µ + w n − + 2( w n + w n − ) + w n +1 dw n − w n dw n −∗ −∗ −∗ . Asterisked entries are placed to avoid the repetitions of entries, since a Poisson-bracket isskew-symmetric J i,j = − J j,i . One can check that the invariants (65) are in involution with respect to the Poissonbracket (67). Therefore, the map (P.iv) is Liouville integrable.The dual map [ x ] ϕ ∨ iv ([ x ]) = [ x ′ ] has the following components:(Q.iv) x ′ = (cid:2) z + ( y + u − νt ) z + x ( νt − x − y ) (cid:3) β + tα ( z − x ) y ′ = x β, z ′ = xyβ, u ′ = xβz, t ′ = tβx. This map depends on three parameters α, β , and ν . The parameter ν is sharedwith the main map (P.iv). The map (Q.iv) has the following degrees of iterates:(68) { d n } Q.iv = 1 , , , , , , , , , , , , , , , , , , , , . . . with generating function:(69) g Q.iv ( s ) = − s + s + s + 1( s + 1)( s + 1)( s − . This means that the dual map is integrable according to the algebraic entropy testwith quadratic growth, just like the main map.Since the main map (P.iv) possesses two invariants and depends on a, b and d whereas the dual map (Q.iv) do not depend on them according to (32) we can writedown the invariants for the dual map (Q.iv) as:(70) αI P.ivlow + βI P.ivhigh = aI Q.iv + dI Q.iv + bI Q.iv . Therefore we obtain the following expressions: t I Q.iv = ( y + z )( α − νβ ) t + ( y + z + 2 yz + uz + xy ) β, (71a) t I Q.iv = νyz ( y + z ) ( α − νβ ) t + (cid:2) yz ( uy + xz + 2 ux ) βν − yz ( ux − xy − yz − uz − y − z ) α (cid:3) t + yz ( x + u + z + y )( y + z + 2 yz + uz + xy ) β, (71b) t I Q.iv = yz ( α − νβ ) t + yz ( x + u + z + y ) β. (71c)The invariants (71a) and (71c) both have degree pattern (1 , , , . However, thesecond invariant is not minimal and it can be replaced with an invariant of degreepattern (1 , , , . Moreover, using the test of [20], we obtain that the map (Q.iv)is not variational. Therefore, we conclude that the dual map (Q.iv) is integrable inthe sense of the existence of invariants, i.e. according to definition 5.4.5. Maps (v).
The main map [ x ] ϕ v ([ x ]) = [ x ′ ] has the following components:(P.v) x ′ = − d ( x + z ) y − (cid:2) ν ( x + z ) t + uz (cid:3) dy − ct y − t a,y ′ = dx y , z ′ = dy x , u ′ = dzx y , t ′ = dtx y . This map depends on the parameters a, c, d and ν . The map (P.v) has the followingdegrees of iterates:(72) { d n } P.v = 1 , , , , , , , , , , , , , . . . with generating function:(73) g P.v ( s ) = − s + 3 s + 4 s − s − s + 2 s ( s − . I-RATIONAL MAPPINGS IN 4D 19
This means that the main map is integrable according to the algebraic entropy testwith quadratic growth.The map (P.v) has the following invariants: t I P.vlow = z d ( x + z ) y + z d (cid:0) νt + zu − ux (cid:1) y + t ( cz + at ) y + t az (74a) t I P.vhigh = z d ( x + z ) y + 2 z ud ( x + z ) y + (cid:8) z du + (cid:2)(cid:0) c − ν d (cid:1) t + 2 xuνd (cid:3) t z + ( at + xc ) t z + t ax (cid:9) y + (cid:2) ( at + uc ) z − t νcz − aνt (cid:3) t y + zt a (cid:0) zu − νt (cid:1) . (74b)Using the methods of [20] we have that the map (P.v) is variational. In affinecoordinates w n its Lagrangian is given by:(75) L P.v = w n w n +1 w n +2 + w n +1 w n νw n +1 w n − ad w n + cd log ( w n ) Using Corollary 2 we obtain the following non-degenerate Poisson structure(76) J P.v = w n − − w n w n − + w n w n +1 + w n − w n − ) + νw n − w n w n −∗ −∗ −∗ . One can check that the invariants (74) are in involution with respect to the Poissonbracket (76). Hence, the map (P.v) is Liouville integrable.The dual map [ x ] ϕ ∨ v ([ x ]) = [ x ′ ] has the following components:(Q.v) x ′ = (cid:2) ν ( x − z ) t + ( u + y ) z − x y (cid:3) β − t α ( x − z ) ,y ′ = βx , z ′ = βx y, u ′ = βx z, t ′ = βx t. This map depends on three parameters α, β and ν . The parameter ν is shared withthe main map (P.v). The map given by (Q.v) has the following degrees of iterates:(77) { d n } Q.v = 1 , , , , , , , , , . . . with generating function:(78) g Q.v ( s ) = − s + 1( s − . This means that the main map is integrable according to the algebraic entropy testwith quadratic growth like the main map.The main map (P.v) possesses two invariants and depends on a, c and d whereasthe dual map (Q.v) do not depend on them. Then according to (32) we can writedown the invariants for the dual map (Q.v) as:(79) αI P.vlow + βI P.vhigh = aI Q.v + cI Q.v + dI Q.v . Therefore we obtain the following expressions: t I Q.v = β (cid:2) ( x + z ) y + yz + uz (cid:3) t + ( α − βν )( y + z ) t , (80a) t I Q.v = (cid:8)(cid:2) ( z + x ) y + zu − νt (cid:3) β + t α (cid:9) zy, (80b) t I Q.v = y z nh ( x + z ) y + 2 uz ( x + z ) y − ν t + 2 νt ux + u z i β + (cid:2) ( x + z ) y + νt − ux + zu (cid:3) t α o . (80c)We note that the degree pattern of these invariants is (1 , , , , (1 , , , and (2 , , , respectively. Finally, using the test of [20], we obtain that the map (Q.v)is not variational. Therefore, we conclude that the dual map (Q.v) is integrable inthe sense of the existence of invariants, i.e. according to definition 5.4.6. Maps (vi).
The main map [ x ] ϕ vi ([ x ]) = [ x ′ ] has the following compo-nents:(P.vi) x ′ = − δat − δ [( u − y ) µaδ + cy + d ( x + z )] t + n aµ h uy + ( x + z ) y + uz i δ + d ( x + z ) y o t − µ h ( x + z ) y + uz i ay y ′ = aµx (cid:0) δt − y (cid:1) (cid:0) δt − x (cid:1) , z ′ = aµy (cid:0) δt − y (cid:1) (cid:0) δt − x (cid:1) ,u ′ = aµz (cid:0) δt − y (cid:1) (cid:0) δt − x (cid:1) , t ′ = aµt (cid:0) δt − y (cid:1) (cid:0) δt − x (cid:1) . This map depends on the five parameters a, c, d and µ, δ . The maps (P.vi) it is aslight generalization of d P II equation which was discussed in [25]. Here, we recallits properties and we discuss its duality in the parameter space. First, the map(P.vi) has the following degrees of iterates:(81) { d n } P.vi = 1 , , , , , , , , , , , , , . . . with generating function:(82) g P.vi ( s ) = − s + 3 s + 4 s − s − s + 2 s ( s − . This means that the main map is integrable according to the algebraic entropy testwith quadratic growth.The map (P.vi) has the following invariants: t I P.vilow = aδ ( y + z ) t − (cid:2) ( u ( x − z ) − xy ) µaδ − cyz − d (cid:0) y + z (cid:1)(cid:3) δt − (cid:8) dy z + aδµ (cid:0) y + z (cid:1) [( x + z ) y − ( x − z ) u ] (cid:9) t + aµy z [( x + z ) y − ( x − z ) u ] (83a) I-RATIONAL MAPPINGS IN 4D 21 t I P.vihigh = δ a ( u + x + y + z ) t + δ (cid:2) aδµ (cid:0) u − uy + x − xz + y + z (cid:1) + ( cu + dx + dz ) y + ( cx + du ) z + xdu ] t − δa (cid:2) ( x + z ) y + yz + uz (cid:3) t − δ (cid:8)(cid:2)(cid:0) u + 2 x + xz + z (cid:1) y + z (2 x + z ) uy + z (cid:0) u + x (cid:1)(cid:3) µδa + d ( x + z ) y + ( x + z ) ( cz + du ) y + z ( cu + dx + dz ) y + duz ( x + z ) (cid:9) t + (cid:26) µaδ (cid:20)
12 ( x + z ) y + uz ( x + z ) y + u z z (cid:18) u + x + xz + z (cid:19) y + uz ( x + z ) y (cid:21) + dy z ( x + z ) ( u + y ) (cid:27) t − [( x + z ) y + uz ] µaz y (83b)Using the methods of [20] we have that the map (P.vi) is variational. In affinecoordinates w n its Lagrangian is given by:(84) L P.vi = (cid:0) w n +1 − δ (cid:1) w n w n +2 + w n +1 w n − daµ w n +1 w n − aµ (cid:20) δ ( δaµ − c ) log (cid:0) w n − δ (cid:1) + 2 a √ δ arctanh (cid:18) w n √ δ (cid:19)(cid:21) Using Corollary 2, we obtain the following non-degenerate Poisson structure for themap (P.vi)(85) J P.vi = w n − − δ − aµ ( w n w n − + w n w n +1 w n − w n − ) − daµ (cid:0) δ − w n − (cid:1) ( δ − w n )0 0 0 1 w n − δ −∗ −∗ −∗ . One can check that the invariants (83) are in involution with respect to the Poissonbracket (85). Therefore, the map (P.vi) is Liouville integrable.The dual map [ x ] ϕ ∨ vi ([ x ]) = [ x ′ ] has the following components:(Q.vi) x ′ = (cid:2) δ t u − ( y + u ) z + x y (cid:3) β + α t ( x − z ) y ′ = βx (cid:0) δ t − x (cid:1) , z ′ = βy (cid:0) δ t − x (cid:1) ,u ′ = βz (cid:0) δ t − x (cid:1) , t ′ = βt (cid:0) δ t − x (cid:1) . This map depends on three parameters α, β and δ . The parameter δ is shared withthe main map (P.vi). The map given by (Q.vi) has the following degrees of iterates:(86) { d n } Q.vi = 1 , , , , , , , . . . with generating function:(87) g Q.vi ( s ) = − s + 1( s − . This means that the main map is integrable according to the algebraic entropy testwith quadratic growth like the main map.The main map (P.vi) possesses two invariants and depends on a, c and d whereasthe dual map (Q.vi) do not depend on them. Then according to (32) we can writedown the invariants for the dual map (Q.vi) as:(88) αI P.vilow + βI P.vihigh = aI Q.vi + cI Q.vi + dI Q.vi . Therefore we obtain the following expressions: t I Q.vi = δ [ β ( u + x + y + z ) δ − α ( y + z )] t + δ µ (cid:8) β (cid:0) u − uy + x − xz + y + z (cid:1) δ + [ u ( x − z ) − xy ] α } t − βδ (cid:2) ( x + z ) y + yz + uz (cid:3) t − δµ (cid:8) βδ (cid:2)(cid:0) u + 2 x + xz + z (cid:1) y + z (2 x + z ) uy + z (cid:0) u + x (cid:1)(cid:3) + α (cid:0) y + z (cid:1) [ u ( x − z ) − ( x + z ) y ] (cid:9) t + 2 µ (cid:26) βδ (cid:20) y x + z ) + uz ( x + z ) y + uz ( x + z ) y + z (cid:18) u + x + xz + z (cid:19) y + u z (cid:21) + αy z u ( x − z ) − ( x + z ) y ] (cid:27) t − [( x + z ) y + uz ] βz µy (89a) t I Q.vi = β (cid:2) u (cid:0) δt − z (cid:1) y + t xzδ − z ( x + z ) y (cid:3) − αyzt (89b) t I Q.vi = β (cid:0) δt − z (cid:1) (cid:0) δt − y (cid:1) ( x + z ) ( u + y ) − α (cid:2) δ (cid:0) y + z (cid:1) t − y z (cid:3) t (89c)We note that the degree pattern of these invariants is (2 , , , , (1 , , , and (1 , , , respectively. Finally, using the test of [20], we obtain that the map (Q.vi)is not variational. Therefore, we conclude that the dual map (Q.vi) is integrable inthe sense of the existence of invariants, i.e. according to definition 5.5. Summary and outlook
In this paper we presented the (P,Q) class of four-dimensional maps. These mapswere obtained by assuming they possess two invariants satisfying the conditions A,B and C given in section 3. In section 4 we discussed the integrability propertiesof these maps.Integrability in the (P,Q) list can arise in different ways depending weather themap is variational or not. Variational maps are all Liovuille integrable, as remarkedin section 2. The only additional structure needed for integrability was then theLagrangian, constructed using the method in [20]. On the other hand integrability
I-RATIONAL MAPPINGS IN 4D 23 in the non-variational maps can arise in two different ways. The pair of maps (P.i)and (Q.i) possessing cubic growth is deflatable . This means that the two mapsarise as non-invertible non-local transformation from two lower-dimensional maps.In [21] we proved that the invariants are preserved in this process and that theintegrability of the three-dimensional maps can be understood using the definitionof Liouville integrability with a rank two Poisson structure. All the other mapspossess quadratic growth and possess a third invariant of motion. In the case ofthe map (P.iii) the third invariant was found by direct inspection, while in all theother cases it was produced directly from the duality approach. As last remark, wenote that the maps with three invariants admit three different degenerate Poissonstructure constructed using the method of [6], but this construction does not yieldLiouville integrability.All the remaining maps have exponential growth and are therefore non inte-grable in the sense of the algebraic entropy. Direct search of invariants for thesemaps excluded the their existence up to order 14. Moreover, using the test of [20],we proved that these exponentially-growing maps are not variational. Thereforewe have a strong evidence of the fact that these maps do not possess any non-degenerate Poisson structure, and therefore these cannot be Liouville integrable.Unfortunately, this result is not enough for a complete proof of the fact that nonon-degenerate Poisson structure exists at all. This is because, in principle, afourth-order recurrence relation can be cast into a system of two second-order re-currence relations which can be variational. Therefore, as we did in [21], we con-jecture that the maps (P.ii) and (Q.iii) either do not admit any full-rank Poissonstructure, or for all full-rank Poisson structure they admit their invariants do notcommute.In table 1 we give a schematic resume of all the above considerations.Equation Degree pattern of invariants Degree of growth Variational(P.i) * (1,3,3,1), (2,4,4,2) cubic no(Q.i) * (1,2,2,1), (2,4,4,2) cubic no(P.ii) (1,2,2,1), (1,3,3,1) exponential no(Q.ii) (2,4,4,2) exponential no(P.iii) (1,3,3,1), (2,4,4,2), (2,5,5,2) quadratic no(Q.iii) (1,2,2,1), (2,4,4,2) exponential no(P.iv) ( d P (2) I ) (1,3,3,1), (2,4,4,2) quadratic yes(Q.iv) (1,2,2,1), (1,2,2,1), (2,4,4,2) quadratic no(P.v) (1,3,3,1), (2,4,4,2) quadratic yes(Q.v) (1,2,2,1), (1,2,2,1), (2,4,4,2) quadratic no(P.vi) ( d P (2) II ) (1,3,3,1), (2,4,4,2) quadratic yes(Q.vi) (1,2,2,1), (1,2,2,1), (2,4,4,2) quadratic no * Deflatable to a three-dimensional Liouville integrable map [21].
Table 1.
Integrability properties of the (P,Q) maps.The search procedure carried out in this paper has been very fruitful givingsome interesting and non-trivial examples of four-dimensional maps. Indeed, all the maps, but four are new. Particularly interesting is the variety of behaviours weencountered in the maps of the class (P,Q). Work is in progress to characterize thesurfaces generated by the invariants in both integrable and non-integrable cases. Weexpect this to give some hints on how the integrability arises from purely geometricalconsiderations. This is well known for maps in two dimension with the theory ofelliptic fibrations applied to the QRT mapping [13]. However, it was discussedin [21, 25] how examples with cubic growth can go beyond the existence of ellipticfibrations making the underlying geometrical structure more complex and richier.Finally, we believe that the direct search of maps with invariants alongside withthe algorithmic tests available in the discrete setting may produce many new resultsand integrable maps in the next years. Analogous procedure in the continuous casestill yield many new result after more that fifty years of their introduction [14,16,32].Work is in progress to extend the present class by considering invariants of moregeneral form. 6.
Acknowledgments
This research was supported by an Australian Laureate Fellowship
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School of Mathematics and Statistics F07, The University of Sydney, NSW 2006,Australia
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