Complexity and integrability in 4D bi-rational maps with two invariants
CCOMPLEXITY AND INTEGRABILITY IN 4D BI-RATIONALMAPS WITH TWO INVARIANTS
GIORGIO GUBBIOTTI, NALINI JOSHI, DINH THI TRAN,AND CLAUDE-MICHEL VIALLET
Abstract.
In this letter we give fourth-order autonomous recurrence rela-tions with two invariants, whose degree growth is cubic or exponential. Theseexamples contradict the common belief that maps with sufficiently many in-variants can have at most quadratic growth. Cubic growth may reflect theexistence of non-elliptic fibrations of invariants, whereas we conjecture thatthe exponentially growing cases lack the necessary conditions for the applica-bility of the discrete Liouville theorem. Introduction
Bi-rational maps in two dimensions have played a crucial role in the study ofintegrable discrete dynamical systems since the seminal paper of [33] and the in-troduction of the QRT mappings in [37, 38]. Elliptic curves and rational ellipticsurfaces proved to be one of the main tools in understanding the geometry behindthis kind of integrability, see [12, 41, 43]. In this letter we give examples of higher-order maps whose properties go beyond those of the two-dimensional maps, andshow that the geometry of elliptic fibrations is no longer sufficient to explain theirbehaviour.Up to now the QRT mappings appear to describe almost the totality of theknown integrable examples in dimension two with some notable exceptions [12, 47].However, no general framework exists for higher order maps. A generalization ofthe QRT scheme [37, 38] in dimension four was given in [7]. Certain maps obtainedin [7] were shown in [19] to be autonomous reductions of members of q -Painlevéhierarchies ( multiplicative equations in Sakai’s scheme [41]). Since hierarchies areknown also for the additive discrete Painlevé equations [10], it is clear that the casesconsidered in [7] cannot exhaust all the possible integrable autonomous maps in fourdimensions, as already shown in [22]. It is important to mention that there are alsoother examples of discrete mappings of higher orders produced either by periodicor symmetry reductions of integrable partial difference equations [27, 32, 36, 44] oras Kahan-Hirota-Kimura discretization [23, 24] of continuous integrable systems[8, 9, 34, 35].In this letter, we focus on the study of integrability properties of autonomousrecurrence relations. Here an autonomous recurrence relation is given by bi-rationalmap of the complex projective space into itself:(1) ϕ : [ x ] ∈ CP n → [ x (cid:48) ] ∈ CP n , Date : May 31, 2019.2010
Mathematics Subject Classification. a r X i v : . [ n li n . S I] M a y G. GUBBIOTTI, N. JOSHI, D. T. TRAN, AND C-M. VIALLET where n > . We take [ x ] = [ x : x : · · · : x n +1 ] and [ x (cid:48) ] = (cid:2) x (cid:48) : x (cid:48) : · · · : x (cid:48) n +1 (cid:3) tobe homogeneous coordinates on CP n . Moreover we recall that a bi-rational map isa rational map ϕ : V → W of algebraic varieties V and W such that there exists amap ψ : W → V , which is the inverse of ϕ in the dense subset where both mapsare defined [42].Integrability for autonomous recurrence relations (discrete equations) can becharacterized in different ways. In the continuous case, for finite dimensional sys-tems, integrability is usually understood as the existence of a “sufficiently” highnumber of first integrals , i.e. of non-trivial functions constant along the solutionof the differential system. In the Hamiltonian setting a characterization of inte-grability was given by Liouville [28]. In the case of maps (1) the analogue of firstintegrals are the invariants . To be more precise we state the following: 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 left unaltered by action of the map, i.e.(2) 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:(3) ∂I∂x ∂I∂x n (cid:54) = 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.A polynomial invariant in the sense of definition 2 written in homogeneous vari-ables is always a rational function homogeneous of degree 0. The form of thepolynomial invariant in homogeneous coordinates is then given by:(4) I ([ x ]) = I (cid:48) ([ x ]) t d , d = deg I (cid:48) ([ 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:(5) dp F = (cid:0) deg x F, deg x F, . . . , deg x n F (cid:1) . Remark . The degree pattern of a polynomial function F is not invariant undergeneral bi-rational transformations. However, the degree pattern of a polynomialfunction F is invariant under scaling and translations, which are transformationsof the form:(6) χ : [ x ] → [ a x + b ] , a ∈ C \ { } , b ∈ CP n . Bi-rational maps in CP are just Möbius transformations so everything is trivial. OMPLEXITY & INTEGRABILITY IN 4D BI-RATIONAL MAPS WITH 2 INVARIANTS 3
Example . Consider the following map in CP :(7) ϕ : [ x : y : t ] (cid:55)→ [ − y ( x − t ) + 2 axt : x ( x − t ) : t ( x − t )] . This map is known as the
McMillan map [30] and possesses the following invariant:(8) t I McM = x y + ( x + y − axy ) t . We have dp I McM = (2 , , i.e. it is a bi-quadratic polynomial. We also note thatthe invariant of a QRT map [37,38], I QRT , which is a generalization of the McMillanmap (7), is the ratio of two bi-quadratics in the dynamical variables of CP . HenceQRT mappings leave invariant a pencil of curves of degree pattern (2 , . Example . The invariants of the maps presented in [7], I CS , are are ratios of bi-quadratics in all the four dynamical variables of CP , i.e. ratios of polynomial ofdegree pattern (2 , , , . In this sense the classification of [7] is an extension ofthe one in [37, 38].Finally we will consider invariants are not of the most general kind, but satisfythe following condition. Definition 4.
We say that a invariant I : CP n → C is symmetric if it is leftunaltered by 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 .We then have the following characterization of integrability for autonomous re-currence relations:(i) Existence of invariants A n -dimensional map is (super)integrable if thereexists n − invariants .(ii) Liouville integrability [4, 29, 45] A n -dimensional map (in affine coordi-nates) is integrable if it preserves a Poisson structure of rank r and r + n − r = n − r functionally independent invariants in involution with re-spect to this Poisson structure . In affine coordinates w = ( w n − , . . . , w ) =[ w n − : · · · : w : 1] we say that a map ϕ : w (cid:55)→ w (cid:48) is called a Poisson map ofrank r ≤ n if there is a skew-symmetric matrix J ( w ) of rank r satisfies theJacobian identity(10) n (cid:88) 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. and(11) d ϕJ ( w ) d ϕ T = J ( w (cid:48) ) , where d ϕ is the Jacobian matrix of the map ϕ , see [7,31]. The Poisson bracketof two smooth functions f and g is defined as(12) { f, g } = ∇ f J ( ∇ g ) T , where ∇ f is the gradient of f . We can easily see that { w i − , w j − } = J ij . Wenote that in the case where the Poisson structure has full rank, i.e. n = 2 r ,we only need n/ invariants which are in involution. In this case the Poissonmatrix is invertible, and its inverse is called a symplectic matrix . A symplecticmatrix give raise to a sympectic structure . G. GUBBIOTTI, N. JOSHI, D. T. TRAN, AND C-M. VIALLET (iii)
Existence of a Lax pair [26] An n -dimensional map is integrable if it arisesas compatibility condition of an overdetermined linear system . We emphasizethe fact that the Lax pair needs to provide us some integrability aspects ofthe maps such as invariants or solutions of the non-linear system. It is knownin the literature that not all the Lax pairs satisfy such conditions [6,18,20,21].Lax pairs that do not satisfy such conditions are called fake Lax pairs andtheir existence cannot be used to prove integrability of a given system.(iv) Low growth condition [2, 13, 46] An n -dimensional bi-rational map isintegrable if the degree of growth of the iterated map ϕ k is polynomial withrespect to the initial conditions [ x ] . Integrability is then equivalent to thevanishing of the algebraic entropy :(13) ε = lim k →∞ k log deg [ x ] ϕ k . Algebraic entropy is a measure of the complexity of a map, analogous to theone introduced by Arnol’d [1] for diffeomorphisms. In this sense growth isgiven by computing the number of intersections of the successive images of astraight line with a generic hyperplane in complex projective space [46].We emphasize the fact that the above list is not completely exhaustive of all thepossible definitions of integrability. Since we are focused on autonomous recurrencerelations we choose to cover only the most used definition for these ones.
Remark . We note that algebraic entropy is invariant under bi-rational maps[15]. In principle, the definition of algebraic entropy in equation (13) requiresus to compute all the iterates of a bi-rational map ϕ to obtain the sequence (cid:110) d k = deg [ x ] ϕ k (cid:111) ∞ k =0 . Fortunately, for the majority of applications the form ofthe sequence can be inferred by using generating functions [25]:(14) g ( z ) = ∞ (cid:88) 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. When a generating function is available, the alge-braic entropy is then given by the logarithm of the smallest pole of the generatingfunction, see [15, 16].
Remark . The condition of Liouville integrability [4, 29, 45] is stronger than theexistence of invariants. Indeed, for a map, being measure preserving and preservinga Poisson/symplectic structure are very strong conditions. However, they lead toa great drop in the number of invariants needed for integrability. The same can besaid for the existence of a Lax pair, since it is well known that a well posed Lax pairgives all the invariants of the system through the spectral relations. Finally, the lowgrowth condition means that the complexity of the map is very low, and it is knownthat invariants help in reducing the complexity of a map. Indeed the growth of amap possessing invariants cannot be generic since the motion is constrained to takeplace on the intersection of hypersurfaces defined by the invariants. For maps in CP , it was proved in [11] that the growth can be only bounded, linear, quadratic orexponential. Linear cases are trivially integrable in the sense of invariants. We notethat for polynomial maps, it was already known from [46] that the growth can beonly linear or exponential. It is known that QRT mappings and other maps with OMPLEXITY & INTEGRABILITY IN 4D BI-RATIONAL MAPS WITH 2 INVARIANTS 5 invariants in CP possess quadratic growth [12], so the two notions are actuallyequivalent for a large class of integrable systems.Now we discuss briefly the concept of duality for rational maps, which was intro-duced in [39]. Let us assume that our map ϕ possesses L independent invariants,i.e. I j for j ∈ { , . . . , L } . Then we can form the linear combination:(15) H = α I + · · · + α L I L . For an unspecified autonomous recurrence relation(16) [ x : x : · · · : x n +1 ] (cid:55)→ [ x (cid:48) : x : · · · : x n ] we can write down the invariant condition for H (15):(17) (cid:98) H ( x (cid:48) , [ x ]) = H ([ x (cid:48) ]) − H ([ x ]) = 0 . Since we know that [ x (cid:48) ] = ϕ ([ x ]) is a solution of (17) we have the following factor-ization:(18) (cid:98) H ( x (cid:48) , [ x ]) = A ( x (cid:48) , [ x ]) B ( x (cid:48) , [ x ]) . We can assume without loss of generality that the map ϕ corresponds to the an-nihilation of A in (18). Now since deg x (cid:48) (cid:98) H = deg x H and deg x n (cid:98) H = deg x n H we have that if if deg x H, deg x n H > the factor B in (18) is non constant . Ingeneral, since the map ϕ is bi-rational, we have the following equalities: deg B x (cid:48) = deg x (cid:48) (cid:98) H − deg x (cid:48) A = deg x H − , (19a) deg B x n = deg x n (cid:98) H − deg x n A = deg x n H − . (19b)Therefore we have that if deg x H, deg x n H > , the annihilation of B doesnot define a bi-rational map in general, 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:(20) (cid:98) H ( x (cid:48) , [ x ]) = d (cid:89) i =1 A i ( x (cid:48) , [ x ]) , but we will not consider this case here.Now assume that the invariants (and hence the map ϕ ) depends on some arbitraryconstants I i = I i ([ x ]; a i ) , for i = 1 , . . . , M . 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 (15) in the following way:(21) 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. The parameters a i k do notappear in the dual maps in the same way as the parameters α i do not appear in themain maps. Therefore, using the factorization (18) the J i functions are invariantsfor the dual maps. 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.
G. GUBBIOTTI, N. JOSHI, D. T. TRAN, AND C-M. VIALLET
Remark . In fact, one can consider more general combinations than linear combi-nations given in (15) and (21). However, we only consider those linear combinationsgiven (15) and (21) in this paper.It is clear from equation (21) that even though the dual map is naturally equippedwith some invariants, it is not necessarily equipped with a sufficient number ofinvariants to claim integrability. In fact there exist examples of dual maps withany possible behaviour, integrable, superintegrable and non-integrable [17, 22].In a recent paper [22], the authors considered the autonomous limit of the secondmember of the d P I and d P II hierarchies [10]. We will denote these equations as d P (2) I and d P (2) II equations. These d P (2) I and d P (2) II equations are given by autonomousrecurrence relations of order four, and showed to be integrable according to thealgebraic entropy approach. They showed that both maps possess two invariants,one of degree pattern (1 , , , and one of degree pattern (2 , , , . Using theseinvariants, they showed that the dual maps of the d P (2) I and d P (2) II equations areintegrable according to the algebraic entropy test and moreover, produced someinvariants, showing that these dual maps were actually superintegrable. Finallythey gave a scheme to construct autonomous recurrence relations with the assigneddegree pattern (1 , , , associated with I low and (2 , , , associated with I high and they provided some new examples out of this construction.In a forthcoming paper [17] we consider the problem of finding all fourth orderbi-rational maps ϕ : [ x : y : z : u : t ] (cid:55)→ [ x (cid:48) : y (cid:48) : z (cid:48) : u (cid:48) : t (cid:48) ] possessing a polynomiala symmetric invariant I low such that dp I low = (1 , , , where the only non-zerocoefficients are those appearing in the (1 , , , invariant of both the d P (2) I and d P (2) II equation , and such that ϕ possesses a polynomial symmetric invariant I high such that dp I high = (2 , , , . The two invariants I low and I high are assumed tobe functionally independent and non-degenerate. Within this class we have foundthe known d P (2) I and d P (2) II equations as well as new examples of maps with theseproperties.In this letter we will present in detail four particular examples of this class. InSection 2, we will discuss two pairs of main-dual maps. We will discuss the integra-bility property of these maps in light of their invariants and of their growth. Wewill present maps possessing two invariants and integrable according to the alge-braic entropy test with cubic growth . This implies that another rational invariantcannot exist. Indeed, the orbits of superintegrable maps with rational invariant areconfined to elliptic curves and the growth is at most quadratic [3, 14]. From thisgeneral statement follows that a four-dimensional map with cubic growth can pos-sess at most two rational invariants. We note that some examples of cubic growthwere already presented in [22]. However, it was pointed out that these examplescan be deflated to lower dimensional maps with quadratic growth. This also holdsfor our maps, i.e. we can deflate them to integrable maps in lower dimension. Fur-thermore, we will present a map with two invariants and exponential growth , thatis non-integrable according to the algebraic entropy test. We discuss some possiblereasons why this map is non-integrable even though it possesses two invariants.In the final Section, we will give some conclusions and an outlook on the futureperspectives of this approach. OMPLEXITY & INTEGRABILITY IN 4D BI-RATIONAL MAPS WITH 2 INVARIANTS 7 Notable examples
In this section we discuss two pairs of maps, which arise as part of a systematicclassification to be presented in [17]. The interest in these particular maps arisessince the relation between their invariants and growth properties is non trivial.In both cases the main maps possess two functionally independent invariants, butthey behave differently. One map has cubic degree growth, while the other one has exponential degree growth. Therefore, even though these two maps have the samenumber of invariants with the same degree patterns, one map is integrable and theother one is non-integrable. In addition, in both cases the degree growth propertyof the dual maps reflect the growth of the main map. However, we note that thedegree growth of the dual map does not always reflect that of the main map [17].2.1. (P.i) and its dual map (Q.i) . Consider the map [ x ] (cid:55)→ ϕ i ([ x ]) = [ x (cid:48) ] givenas follows:(P.i) x (cid:48) = −{ [ νt ( x + z ) + uz ] y + t µuz + ( x + z ) y } d − at ,y (cid:48) = x d ( t µ + xy ) , z (cid:48) = yxd ( t µ + xy ) ,u (cid:48) = zxd ( t µ + xy ) , t (cid:48) = txd ( t µ + xy ) . This map depends on four parameters a, d and µ, ν .From the construction in [17] we know that the map (P.i) possesses the followinginvariants: 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 ) , (22a) t I P.ihigh = [( uz + xy − yz ) µ − νyz ] at + (cid:2) yz ( xy + yz + uz ) a + dµ ( uz + xy − yz ) + 2 dµνyz ( ux − yz ) − dν y z (cid:3) t + (cid:2) dzy ( uz + xy − yz )( xy + yz + uz ) µ + 2 dy z νux (cid:3) t + dy z ( xy + yz + uz ) . (22b)Moreover, the map (P.i) has the following degrees of iterates:(23) { d n } P.i =1 , , , , , , , , , , , , , , , , , , . . . The generating function of the sequence (23) is given by:(24) g P.i ( s ) = s − s + s − s + 3 s + 3 s + s + 1( s + 1)( s + 1)( s − . Due to the presence of ( s − in the denominator we have that the growth of themap (P.i) is fitted by a cubic polynomial . As discussed in the Introduction thismeans at once that the map is integrable according to the algebraic entropy testand that another rational invariant cannot exist. This suggests that the geometryof the orbits of the map (P.i) is nontrivial, and goes beyond the existence of ellipticfibrations .Explicit numerical calculations and drawings suggest that in the case of map(P.i), no additional invariant exists. Indeed, if an additional third invariant, evenalgebraic, existed then all the orbits of of equation (P.i) would lie on a curve. On G. GUBBIOTTI, N. JOSHI, D. T. TRAN, AND C-M. VIALLET the other hand referring to Figure 1 we see that a generic orbit of equation (P.i)does not lie on a curve. This implies that no such an invariant might exist.
Figure 1.
Affine orbit of equation (P.i) with parameters a = 6 , µ = 3 , ν = 4 and d = 6 and initial conditions ( x, y, z, u ) =(0 . , . , . , . .The dual map [ x ] (cid:55)→ ϕ ∨ i ([ x ]) = [ x (cid:48) ] of (P.i) is given by:(Q.i) x (cid:48) = [ β (2 xy − yz + uz ) µ + ( βν − α ) y ( x − z )] t + βy ( z y − x y + uz ) y (cid:48) = x β ( t µ + xy ) , z (cid:48) = yxβ ( t µ + xy ) ,u (cid:48) = zxβ ( t µ + xy ) , t (cid:48) = txβ ( t µ + xy ) . This map depends on three parameters α, β , and µ, ν . The parameters µ and ν areshared with the main map (P.i).The main map (P.i) possesses two invariants and depends on a and d whereasthe dual map (Q.i) does not depend on them. Then according to (21) we can writedown the invariants for the dual map (Q.i) as:(25) 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 ) , (26a) 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. (26b) OMPLEXITY & INTEGRABILITY IN 4D BI-RATIONAL MAPS WITH 2 INVARIANTS 9
We remark that the invariant (26a) has degree pattern (1 , , , which differs from dp I P . i low .The map (Q.i) has the following degrees of iterates:(27) { d n } Q.i = 1 , , , , , , , , , , , , , . . . with generating function:(28) 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.Explicit numerical calculations and drawings suggest that also in the case of map(Q.i), no additional invariant exists. Indeed, if an additional third invariant, evenalgebraic, existed then all the orbits of of equation (Q.i) would lie on a curve. Inthis case we are actually able to find some orbits lying on a curve, see Figure 2b.However, it is possible to find orbits of equation (Q.i) that do not lie on a curve.An example of such orbit is shown in Figure 2a. Therefore, we can conclude that a globally defined third invariant does not exist. The existence of some closed orbitslike in Figure 2b suggest the existence of a non-analytic invariant existing only insome regions of the space. (a)
Parameters α = 3 , µ = 3 , ν = 7 and β = 3 . (b) Parameters α = 3 , µ = 6 , ν = 8 and β = 9 . Figure 2.
Affine orbit of equation (Q.i) with different parametersbut the same initial conditions ( x, y, z, u ) = (3 , , , .Therefore, the pair of main-dual maps (P.i) and (Q.i) consists of two integrableequations with non-standard degree of growth. However, as remarked above thedegree pattern of the invariants of the maps (P.i) and (Q.i) differ slightly.We now consider the maps (P.i) and (Q.i) in affine coordinates which are givenby(29) ϕ : ( w , w , w , w ) (cid:55)→ ( w , w , w , w ) , where w = N dw ( w w + µ ) , (AP.i) w = N β w ( w w + µ ) , (AD.i)with N = − d (cid:0) w w w + w w + 2 w w w + w w + µw w + νw w + νw w (cid:1) − a, (30) N = βw w w + βµw w + βw w + ( α − βµ − βν ) w w − βw w + (2 βµ + βν − α ) w w . (31)Invariants for these maps are obtained from I low and I high respectively by taking t = 1 , u = w , z = w , y = w , and x = w .We note that when a Poisson structure has the full rank, using equation (11),one gets(32) (cid:2) det(d ϕ ) (cid:3) = det (cid:0) J ( w (cid:48) ) (cid:1) det (cid:0) J ( w ) (cid:1) . This implies that the map ϕ is either volume or anti-volume preserving.We recall that a map ϕ is called (anti)volume preserving if there is a function Ω( w ) such that the following volume form is preserved(33) Ω( w ) d w ∧ d w ∧ . . . ∧ d w n − = ± Ω( w (cid:48) ) d w (cid:48) ∧ d w (cid:48) ∧ . . . ∧ d w (cid:48) n − . Thus, we can write(34) ∂ (cid:0) w (cid:48) , w (cid:48) , . . . , w (cid:48) n − (cid:1) ∂ (cid:0) w , w , . . . , w n − (cid:1) = ± Ω( w )Ω( w (cid:48) ) , where the left hand side is the determinant of the Jacobian matrix of the map ϕ .In [5] it was proved that if a map in n dimension is (anti) volume preserving andpossesses n − invariant, then we can construct an (anti) Poisson structure of rank 2from these invariants. However, these invariants turn out to be Casimirs (functionsthat Poisson commute with all other functions) of this Poisson bracket. Therefore,in order to have Liouville integrability we need an extra invariant apart from theknown n − invariants if we want to use use Poisson structures constructed thisway. In other words, the map is super integrable. Thus, to discuss about Liouvilleintegrability of the maps (AP.i) and (AQ.i) we need to find a Poisson bracket of rank as we already predicted that the third invariant does not exist. We do not havethat information for these maps but we can show they reduce to three dimensionalLiouville integrable maps via a process called deflation [22]. Mutatis mutandis ,this process will preserve the invariants, and in dimension three two invariants aresufficient to claim integrability in the general sense as discussed in the Introduction.It is easy to check that the maps (AP.i) and (AQ.i) are volume and anti-volumepreserving, respectively, with respect to the same volume form:(35)
Ω = w w ( w w + µ ) . OMPLEXITY & INTEGRABILITY IN 4D BI-RATIONAL MAPS WITH 2 INVARIANTS 11
We now construct the (anti) Poisson structures for these two maps following [5].We consider the dual multi-vector of the volume form(36) τ = m ∂w ∧ ∂w ∧ ∂w ∧ ∂w , where m = 1 / Ω . A degenerate Poisson structure for the map (AP.i) and a degener-ate anti-Poisson structure for the map (AQ.i) are given by the following contraction(37) J = τ (cid:99) d I low (cid:99) d I high , where I low and I high are invariants for these maps in affine coordinates. Since these(anti) Poisson structures are quite big, we do not present them here. Remark . The Poisson structures which can be constructed using the methodof [5] are degenerate and cannot be used to explain the integrability of the twomaps (AP.i) and (AQ.i).We also note that the maps (AP.i) and (AQ.i) can be reduced to three dimen-sional maps using a deflation v i = w i w i +1 . The recurrences for these maps aredenoted by (DP.i) and (DQ.i) and are given as follows dµ ( v + v ) + dν ( v + v ) + d (cid:0) v v + v + 2 v v + v + v v (cid:1) + a = 0 , (DP.i) βµ ( − v + 2 βv − βv + v ) + ( βν − α ) ( v − v ) (DQ.i) + β (cid:0) − v v − v + v + v v (cid:1) = 0 . Each of the maps (DP.i) and (DQ.i) has two functionally independent invariantswhich can be obtained directly from I low and I high even though they live in adifferent space. One can check that the map (DP.i) and (DQ.i) are anti-volumepreserving and volume preserving with Ω = v + µ . Therefore, we can constructtheir (anti) Poisson structure using the three dimensional version of (37). Usingthe following invariant from I low for (DP.i)(38) I P . i1 = dµv v − dµv v + dµv v + dνv + dv v − dv v v + dv + dv v + av we have found that the map (dP.i) has an anti-Poisson structure given by J P . i12 = d ( v − v ) , J P . i2 , = d ( v − v ) J P . i13 = − dµv − dµv − dνv − dv v + dv v − dv − dv v − aµ + v . Similarly, for the map (DQ.i) we obtain the invariant(39) I DQ.i = βµv − βµv + βµv − νβv + βv v + βv + βv v + αv , and the corresponding Poisson structure(40) J Q . i = β β ( µ + ν − v − v − v ) − αµ + v − β β − β ( µ + ν − v − v − v ) − αµ + v − β . For these constructions, I P . i1 and I Q . i1 are Casimirs for their associated (anti) Pois-son structures. Their second (anti) Poisson structures can be obtained from theinvariant I high but we do not present here as they are quite big. It is important to note that the (anti) Poisson structures of (AP.i)and (AQ.i)under inflation give us the trivial Poisson structures for (DP.i) and (DQ.i), i.e. J = , where is the zero matrix. On the other hand, from the common factorthat appears in the Poisson structure of (AP.i), we have found that there exists ananti-invariant K P . i for this map, i.e. K P . i ( w ) = − K P . i ( w (cid:48) ) where(41) K P . i = 2 d (cid:0) w w w + w w + w w w + µw w − µw w + µw w + νw w (cid:1) + a. However, K P . i is not independent of I P . i low and I P . i high since we have(42) (cid:0) K P . i (cid:1) − d I P . i high − d νI P . i low = a . Using this anti-invariant, we obtain the following anti-invariant for the map (DP.i)(43) K DP.i = 2 dµv − dµv + 2 dµv + 2 dνv + 2 dv v + 2 dv + 2 dv v + a. Therefore, using this anti-invariant, we get a Poisson structure for (DP.i) as follows(after factoring out a constant term)(44) J P . i2 = µ − ν − v − v − v µ + v − − µ − ν − v − v − v µ + v − . We can check directly that the invariants inherited from the affine map (AP.i) are ininvolution with respect to the Poisson structure (44). In the sense of the definitiongiven in the Introduction, this means that the reduced maps (DP.i) and (DQ.i) areLiouville integrable.
Remark . We notice that we can always use the invariants (38) and (39) to reducethe three dimensional maps (DP.i) and (DQ.i) to two dimensional maps and relatethem to QRT maps. To be more specific we have that the reduced map of (DQ.i)preserves a bi-quadratic curve so that it is of the QRT type. On the other hand,using the anti-invariant, the reduced map of (DP.i) sends a bi-quadratic to anotherbi-quadratic and fits in the framework of [40].2.2. (P.ii) and its dual map (Q.ii) . Consider the map [ x ] (cid:55)→ ϕ ii ([ x ]) = [ x (cid:48) ] givenas follows:(P.ii) x (cid:48) = (cid:2) ( x + z ) y − uz (cid:3) µ − t ( u − y ) ,y (cid:48) = x ( t + µx ) , z (cid:48) = y ( t + µx ) ,u (cid:48) = z ( t + µx ) , t (cid:48) = t ( t + µx ) . This map only depends on the parameter µ .From the construction in [17] we know that the map (P.ii) has the followinginvariants: t I P.iilow = ( x − z ) ( u − y ) (cid:0) t + z µ (cid:1) (cid:0) µy + t (cid:1) , (45a) t I P.iihigh = (cid:104) ( x − z ) y + y z − yz u + u z (cid:105) 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) . (45b) OMPLEXITY & INTEGRABILITY IN 4D BI-RATIONAL MAPS WITH 2 INVARIANTS 13
Moreover, the map (P.ii) has the following degrees of iterates:(46) { d n } P.ii = 1 , , , , , , , , , . . . with generating function:(47) g P.ii ( s ) = 1 + 2 s (2 s − s − . This means that despite the existence of the two invariants (45) the map (P.ii) isnon-integrable according to the algebraic entropy test: its entropy is positive andgiven by ε = log 2 .Therefore we have that the map (P.ii) is an example of non-integrable admittingtwo invariants.Again following [5] we can produce a Poisson structure of rank for (P.ii) as theaffine version of (P.ii) is volume preserving with Ω = (1 + µw )(1 + µw ) , wherewe have taken t = 1 , u = w , z = w , y = w , and x = w . By the construction,the two invariants (45) become Casimir functions for it, so again the existence ofsuch Poisson structure does not imply any form of Liouville integrability. However,we notice that there are common factors appear at every non-zero entries of thisstructure. Thus, we have found the following anti-invariant for the map (P.ii) usingthese common factors(48) K P . ii = (cid:2) µ (cid:0) w w − w w − w w + w w (cid:1) + w − w − w + w (cid:3) × (cid:2) µ (cid:0) w w − w w + w w − w w (cid:1) + w + w − w − w (cid:3) = F F This suggests that we should check each factor of K P.ii to see whether they are(anti) invariants of (Pii). By direct calculation we can see that the first factor F is an anti-invariant and F is an invariant for (P.ii), but they are not functionallyindependent of I low and I high . In fact, their relations are(49) I P . ii high − F + 2 I P . ii low = 0 , and I P . ii high − F − I P . ii low = 0 . Therefore, the map (P.ii) actually has two invariants of degrees (1 , , , and (1 , , , . Nevertheless, despite the existence of such invariants the map (P.ii)is non-integrable in the sense of the algebraic entropy. Remark . We can use F and F to construct an anti-Poisson structure for (P.ii)using the formula (37)(50) J , = − , J , = 1 , J , = − J , = 2 µw ( w − w ) µw + 1 , J , = − µw ( w − w ) µw + 1 J , = − µ w w [4 ( w w − w w + w w ) − w w ] + µ (cid:0) w + w (cid:1) + 1( µw + 1) ( µw + 1) We have checked that F and I P.ii low are in involution with respect to this anti-Poissonstructure. A Poisson structure can be obtained by multiplying this anti-Poissonstructure with the anti-invariant F . The dual map [ x ] (cid:55)→ ϕ ∨ ii ([ x ]) = [ x (cid:48) ] of (P.ii) is given as follows:(Q.ii) x (cid:48) = 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 (cid:48) = αx (cid:0) t + µx (cid:1) , z (cid:48) = αy (cid:0) t + µx (cid:1) ,u (cid:48) = αz (cid:0) t + µx (cid:1) , t (cid:48) = αt (cid:0) t + µx (cid:1) . This map depends on three parameters α, β and B . The parameter µ is sharedwith the main map (P.ii).Since the main map (P.ii) possesses two invariantss depending only on one pa-rameter µ then according to (21) we can write down only a single invariant for thedual map (Q.ii):(51) I Q.ii = αI P.iihigh + βI P.iilow . The invariant (51) has degree pattern (2 , , , .We have then that the dual map (Q.ii) has the following fast-growing degrees ofiterates:(52) { d n } Q.ii = 1 , , , , , , , , , , . . . . The growth (52) is clearly exponential and its generating function is(53) g Q.ii ( s ) = 1 + 2 s (2 s − s − . This confirms that the algebraic entropy is positive and equal to ε = log 2 .This means that the dual map is non-integrable with same rate of growth as themain map. In this case we can show that the map is anti-volume preserving withthe same measure as the main map (P.ii). Moreover, we proved that the map (Q.ii)do not possesses any addition invariant up to order 14. Therefore at the presentstage we cannot construct a Poisson structure using the method of [5].3. Conclusions and outlook
In this letter, we gave some examples of fourth order bi-rational maps with twoinvariants possessing interesting degree growth properties. These examples comefrom our forthcoming classification of all the fourth-order autonomous recurrencerelations possessing two invariants in a given class of degree patterns [17] .The first pair of bi-rational maps is given by the map (P.i) and its dual (Q.i)and consists of integrable maps with cubic growth. The interest in maps withcubic growth arises from geometrical considerations: maps with polynomial buthigher than quadratic growth, can arise only in dimension greater than two [11] andprove, in the case of superintegrable maps, the existence of non-elliptic fibrations ofinvariant varieties [2]. The interest in maps with this type of growth arose recentlyfollowing the examples given in [22] and we expect them to lead to many new andinteresting geometric structures.The second pair of fourth order bi-rational maps given by the map (P.ii) andits dual (Q.ii), consists of non-integrable maps with exponential growth. Thereare various possible reasons why the map (P.ii) is non-integrable despite possessingtwo invariants. To claim integrability with two invariants according to the discreteLiouville theorem [4,29,45] we need to prove that the map has a symplectic structureand that the two invariants commute with respect to this symplectic structure.
OMPLEXITY & INTEGRABILITY IN 4D BI-RATIONAL MAPS WITH 2 INVARIANTS 15
Hence, either the map (P.ii) does not admit any symplectic structure, or the map(P.ii) admits only symplectic structures such that the two invariants (45) do notcommute. Since, usually, from a set of non-commuting invariants it is possible tofind a set of functionally independent commuting invariants we are more leaned toconjecture that equation (P.ii) is devoid of a non-degenerate Poisson structure.Work is in progress to characterize the surfaces generated by the invariants inboth integrable and non-integrable cases. We expect this to give new results in thegeometric theory of integrable systems.
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