THE GROUP OF SYMPLECTIC BIRATIONAL MAPS OF THE PLANE AND THE DYNAMICS OF A FAMILY OF 4D MAPS

. We consider a family of birational maps ϕ k in dimension 4, aris- ing in the context of cluster algebras from a mutation-periodic quiver of period 2. We approach the dynamics of the family ϕ k using Poisson geometry tools, namely the properties of the restrictions of the maps ϕ k and their fourth iterate ϕ (4) k to the symplectic leaves of an appropriate Poisson manifold ( R 4+ ,P ). These restricted maps are shown to belong to a group of symplectic birational maps of the plane which is isomorphic to the semidirect product SL (2 , Z ) (cid:110)R 2 . The study of these restricted maps leads to the conclusion that there are three diﬀerent types of dynamical behaviour for ϕ k characterized by the parameter values k = 1, k = 2 and k ≥ 3.


1.
Introduction. This is a companion paper to our works [2,5] on the dynamics of maps arising in the context of the theory of cluster algebras [6] through the notion of mutation-periodic quivers [9] (a.k.a. cluster maps). We study the main geometric features underpinning the dynamics of a family of (cluster) maps ϕ k in dimension 4, depending on a positive integer parameter k. Although most of the dynamical behaviour of these maps is presented in our unpublished work [4], here we approach their dynamics under a different point of view, aiming at keeping the paper as self contained as possible and at the same time highlighting the main geometric aspects relevant to the dynamics.
We consider the family of maps defined in R 4 + by This family is associated to the 4-node quiver represented in Figure 1, which is mutation-periodic of period 2, and is a particular instance of the quiver in [2, 1, Example 4]. By definition of the maps associated to mutation periodic quivers, the maps ϕ k are birational, that is, rational maps with rational inverse.
We refer the reader interested in mutation-periodic quivers and studies of maps associated to mutation-periodic quivers of period 1 to [9,8,7] and to [11,10] for general aspects of cluster algebras and applications.
We approach the study of the dynamics of the maps ϕ k by realizing that they are maps preserving a Poisson structure P of log-canonical type. This structure is regular on R 4 + , and the leaves of the respective symplectic foliation of R 4 + are semi-algebraic sets of dimension 2. All these symplectic leaves are invariant under the fourth iterate ϕ (4) k of the maps ϕ k with one leaf being invariant under ϕ k . The periodic points of the maps ϕ k are then obtained by studying the restrictions of ϕ k and ϕ (4) k to the (invariant) symplectic leaves. This study provides the full description of the periodic points of the family of maps (1) and enables us to conclude that there are three different types of dynamical behaviour according to the parameter values k = 1, k = 2 and k ≥ 3. The identification of the periodic points of ϕ k in the cases k = 1 and k ≥ 3 is described in Theorem 4.1 and Theorem 4.2 respectively, and for k = 2 it can be found in [3,Theorem 3]. In particular, we show that: (a) ϕ 1 is globally 12-periodic, with a unique fixed point and 2-dimensional semi-algebraic sets of points with minimal periods 4 and 6; (b) ϕ 2 has no periodic points; (c) if k ≥ 3, ϕ k has a unique fixed point and a 2-dimensional semi-algebraic set of points of minimal period 4.
The structure of the paper is as follows. The first section recalls that each map ϕ k is a Poisson map with respect to a Poisson structure P of rank 2. We also show that the respective symplectic foliation of R 4 + is invariant under the fourth iterate of ϕ k (Theorem 2.1). The following section is devoted to the study of the restrictions of ϕ k and of ϕ (4) k to the appropriate (invariant) symplectic leaves of the referred foliation. It is shown that these restrictions are symplectic birational maps of the plane belonging to a group, Γ, which is isomorphic to the semidirect product SL(2, Z) R 2 . We also find normal forms for the maps of this group up to conjugation in GL(2, Z) R 2 . The final section is devoted to the study of the periodic points of the maps ϕ k . This is accomplished by using results of the previous section concerning the group Γ to obtain the periodic points of the restricted maps, from which we determine the periodic points of the maps ϕ k .
2. Reduction to globally-periodic symplectic maps. As shown in [2] any map associated to a mutation-periodic quiver leaves invariant a log-canonical presymplectic form defined by the quiver. In the case of the maps ϕ k , this presymplectic form can be used to obtain reduced two dimensional symplectic maps ϕ k . In what follows we will use an alternative approach by considering a Poisson structure which is invariant under ϕ k . This approach leads precisely to the same reduced maps ϕ k , with the symplectic foliation of the Poisson structure replacing the null foliation of the pre-symplectic form. For more details on the role of the null foliation and symplectic foliations arising in the study of these type of maps we refer to [5].
Although each map ϕ k of the family (1) is defined for x 1 x 2 = 0, throughout the paper we consider its domain of definition to be R 4 + which guarantees that any iterate of these maps is well defined.
As follows from Example 4 in [2] (with r = s = k and t = 0) each map ϕ k is a Poisson map with respect to the Poisson structure where the matrix C = [c ij ] is the skew-symmetric matrix That is, for each k ∈ Z + we have (ϕ k ) * P = P , where (ϕ k ) * denotes the pushforward by ϕ k . The Poisson structure P is known as a log-canonical Poisson structure since it is constant in logarithmic coordinates.
It can easily be checked that [c ij x i x j ] i,j=1,...,4 has null determinant and consequently, in R 4 + , the Poisson tensor has constant rank equal to 2, meaning that P is a regular (degenerate) Poisson structure.
Each map ϕ k is a birational Poisson map and so by Theorem 5.1 in [2] there is a submersion Π k and a mapφ k defined on R 2 + such that That is, one has the commutativity of the following diagram Moreover, the submersion Π k sends R 4 + onto a maximal set of independent Casimirs. Such a set can be easily obtained from a basis of the kernel of the matrix C in (3).
4 is a Casimir (see Lemma 5.2 in [2]). Thus, considering Ker C = v 1 , v 2 with v 1 = (1, −k, 0, 1), v 2 = (0, −1, 1, 0), we take for a maximal set of Casimirs, {x, y}, the rational functions x = x v1 and y = x v2 , that is, The submersions Π k are then given by The components of the maps ϕ k are obtained by computing x • ϕ k and y • ϕ k as functions of x and y. This computation gives Remark 1. We note that each Casimir is invariant under a scaling action of the multiplicative group R 2 + on R 4 + with weights defined by the components of vectors forming a basis of the image of the matrix C in (3). Namely, taking Im C = u 1 , u 2 with u 1 = (0, k, k, k 2 ) and u 2 = (−k, 0, 0, k), this scaling action is defined by However, the maps ϕ k are not invariant (neither equivariant) under this scaling action, which leaves us outside the usual Poisson reduction setting.
Another remark is worth mentioning.

Remark 2.
As the Poisson structure P in (2) is regular in R 4 + all its (2-dimensional) symplectic leaves are the common level sets of a maximal set of independent Casimirs. However, it can be shown that the maps ϕ k do not preserve this symplectic foliation of R 4 + , and therefore none of these maps is the discrete analogue of a Hamiltonian flow.
Taking into account the above remark one might be tempted to consider that the Poisson structure is of no relevance to the study of the dynamics of the family of maps ϕ k . However, as we will show in Theorem 2.1, the fourth iterate of each map ϕ k does preserve the symplectic foliation of R 4 + . Hence, like in the continuous setting, the restriction of these maps to the symplectic leaves can be used to study the dynamics.
Theorem 2.1. The fourth iterate of each map ϕ k in (1) preserves the symplectic foliation of (R 4 + , P ) with P the Poisson structure in (2). That is, where S k (p,q) is a symplectic leaf. In particular, the Casimirsx = x k 2 +x k Proposition 1. Let ϕ k be the maps in (7) defined in R 2 + . Then, (i) each map ϕ k is globally 4-periodic, i.e. ϕ (ii) each map ϕ k is symplectic with respect to the symplectic form A simple computation shows that The parameter independent map ψ is globally 4-periodic (i.e. ψ (4) = Id), and from (8) it follows where the above equivalence comes from the global periodicity of ψ and the fact that h k is a homeomorphism.
(ii) Straightforward computations show that the maps h k and ψ preserve ω, that is the pullback of ω by these maps is ω: The symplectic leaves of (R 4 + , P ) are 2-dimensional subsets of R 4 + (since the rank of P is 2) defined by the common level set of two independent Casimirs of P . These leaves could be defined as the fibres of the submersion Π k in (6) but due to the previous proposition it is more convenient to consider them to be the fibres of We note that, since x and y are Casimirs of P then 1+y k x is also a Casimir and so the components of π k form a maximal set of independent Casimirs. Thus, the symplectic leaves are given by Proof of Theorem 2.1. From (4) and (8) one has with π k the map (10) and ψ as in (9). The last equivalence implies one has π k (x) = (p, q), and again from the last equivalence, . Note that the components of π k are Casimirs of P , therefore the fact that the Casimirs are first integrals of ϕ (4) k is just a consequence of the identity above.
3. Restrictions to symplectic leaves and the group of symplectic birational maps of the plane. The symplectic leaves S k (p,q) defined by (11) are two dimensional semi-algebraic sets invariant under ϕ (4) k . However, there are symplectic leaves which are invariant under a lower order iterate of ϕ k . Indeed, from (4) and (8) one has if and only if (p, q) is an n-periodic point of ψ. A point fixed by the nth iterate of a function and not fixed by any other lower order iterate will be called a point of minimal period n.
In R 2 + , the map ψ has a unique fixed point (1, 1) and any other point is periodic of minimal period 4. So one has the following invariance of the symplectic leaves: k and not invariant under any lower order iterate of ϕ k .
The expressions of the restriction of ϕ k to S k (1,1) and of the restriction of ϕ (4) k to S k (p,q) are given in the following proposition.
Proposition 2. Let ϕ k be the maps (1) and S k (p,q) the 2-dimensional symplectic leaves defined by (11). Then, 1. S k (1,1) is invariant under ϕ k and in the coordinates (x 1 , x 2 ), the restriction is given by: 2. if (p, q) = (1, 1) the symplectic leaves S k (p,q) are invariant under ϕ (4) k and the restriction ϕ k = ϕ is given, in the coordinates (x 1 , x 4 ), by with Proof. As seen previously the invariance properties of the symplectic leaves follows from the type of periodic points of the map ψ in (9). Straightforward computations lead to the expressions of the restricted map in (13). To obtain ϕ k , the computation of ϕ It is easy to see that the function l is constant on each S k (p,q) and given by This leads directly to the expression of ϕ k in the coordinates (x 1 , x 4 ). and for S k (p,q) has no particular meaning other than leading in each case to simpler expressions of the restricted maps.
Using the expressions of the restricted maps (13)-(14) one easily verifies that they are maps of the plane preserving the symplectic form The fact that these maps are symplectic is not a coincidence since: (i) the symplectic structure on a symplectic leaf S is the nondegenerate Poisson structure induced from P , meaning that the inclusion i : S → R 4 + is a Poisson map; (ii) ϕ k is a Poisson map, and so is any iterate ϕ Using algebraic geometry techniques, it was proved by Blanc in [1] that the group of birational transformations of C 2 preserving the symplectic form (17) is generated by SL(2, Z), the complex torus (C * ) 2 and the globally 5-periodic (Lyness) map (x, y) → (y, 1+y x ). Here we consider this group restricted to R 2 + and we will denote it by Γ.
Note that g is the composition of the translation by the vector v = (log α, log β) and an area preserving linear map represented by the SL(2, Z) matrix Identifying g with (M, v), the map i induces an isomorphism between Γ and the semidirect product In order to better identify the type of periodic points of the restricted maps in Proposition 2 we deduce in the next proposition a normal form for all the maps in Γ except the maps f ± (α,β) (x, y) = (αx ±1 , βy ±1 ).
Proposition 3. Let f : R 2 + → R 2 , defined by f (x, y) = αx a y b , βx c y d , ad − bc = 1 α, β = 0 be an element of Γ with b 2 + c 2 = 0. Then f is conjugate to the map: Proof. If c = 0, considering the homeomorphism π given by it is easy to check that π • f = g • π, where g is the map If a + d = 2 the map g is the map f 2,ξ with ξ = α c β a−1 . If a + d = 2, taking the following map Π Π(x, y) = K 1 a+d−2 (x, y), If c = 0, the hypothesis b 2 +c 2 = 0 implies that b = 0. Considering the involution σ(x, y) = (y, x), which interchanges c and b, the problem reduces to the previous cases. In fact, σ • f • σ = βx d , αx b y a is conjugate to f a+d if a + d = 2 and to f 2,ξ As a consequence of the proof of Proposition 3 the restricted maps (13) and (14) are conjugate to the normal forms given in the following corollary. Corollary 1. Let k ∈ Z + and λ be a nonzero real number. Consider the maps 1. If k = 2, then i)φ 2 is already in normal form: Proof. Note that both mapsφ k and ϕ k verify the hypotheses of Proposition 3 with c = 0, for any k. Furthermore, a + d = k forφ k and a + d = (k 2 − 2) 2 − 2 for ϕ k . For both maps a + d = 2 if and only if k = 2. The result then follows from the proof of Proposition 3.
Remark 5. We remark that from the proof of the above corollary the restricted mapsφ k and ϕ k are conjugate to SL(2, Z) maps except in the case k = 2. Moreover: (a) |a + d| = 1 if and only if k = 1, so thatφ 1 and ϕ 1 are conjugate to elliptic SL(2, Z) maps; (b) |a + d| > 2 if and only if k ≥ 3, and soφ k and ϕ k are conjugate to hyperbolic SL(2, Z) maps for k ≥ 3.
For future reference, we now mention the form of an iterate of order n of the maps f 2,ξ (x, y) = (y, ξ y 2 x ), given in Proposition 3-2. This expression can be computed by applying Lemma 1 in [3] or by considering the conjugate affine map This map can be identified with the SL(3, R) matrix and so, computing the nth power of X we arrive at the expression of g (n) 2,ξ from which we obtain: y n x n (x, ξ n y) , n ≥ 0.
4. Periodic points of the maps ϕ k . The existence of periodic points for the maps of the family ϕ k given by (1) is obtained from the periodic points of the maps restricted to the symplectic leaves, namely the mapsφ k and ϕ k in Proposition 2.
In turn, the existence of periodic points of these restricted maps rely on the results of the previous section for the group Γ.

4.1.
Periodic points of the restricted maps. In this subsection we describe the type of periodic points of the restricted maps according to the values of the parameter k.

4.2.
Periodic points of ϕ k . Finally, we address the problem of describing the main dynamical features of the maps of the family (1) defined in R 4 + . Recall that, by Theorem 2.1, R 4 + is foliated by 2-dimensional symplectic leaves S k (p,q) of P (with P as in (2)), all of them invariant under the the fourth iterate ϕ (4) k and with the leaf S k (1,1) invariant under ϕ k . In particular, this means that each orbit of ϕ k is either entirely contained in S k (1,1) or jumps between four pairwise disjoint symplectic leaves all of them invariant under ϕ (4) k . In fact, the leaves S k (p,q) are the fibres of π k and by (12) one has π k • ϕ k = ψ • π k with ψ(x, y) = y, 1 x . So, S k (p,q) ϕ k −→ S k ψ(p,q) . In Theorem 4.1 and Theorem 4.2 below we will characterize the periodic points of ϕ k in the cases k = 1 and k ≥ 3, respectively. The case k = 2 will not be explicitly stated since it is easy to see that the map ϕ 2 has no periodic points and its dynamics is described in detail in our work [3, Theorem 3].
Proof. By Proposition 4-1, the restrictionφ 1 of ϕ 1 to S 1 (1,1) is globally 6-periodic and the restriction ϕ 1 of ϕ (4) 1 to any S 1 (p,q) is globally 3-periodic. Hence ϕ 1 is globally 12-periodic. Moreover, all the points in S 1 (1,1) have minimal period 6 except the point F = (2, 2, 2, 2) which is fixed. Also, any point belonging to S 1 (p,q) is either a fixed point of ϕ 1 , which correspond to periodic points of ϕ 1 with minimal period 4, we note that by Proposition 4-1 and Proposition 2 these are points x ∈ S 1 (p,q) whose coordinates x 1 and x 4 satisfy x 1 = x 4 = λ 1/3 , for λ given by (15) (with k = 1). On the other hand, the constant λ is the value of the restriction to S 1 (p,q) of the function l(x) given in (16). To obtain the set V it is enough to eliminate λ from these relations, that is from Finally, the remaining points are periodic points of ϕ 1 with minimal period 3, and therefore they are periodic points of ϕ 1 with minimal period 12.  (1,1) . 2. a semi-algebraic set V ⊂ S k (p,q) , with (p, q) = (1, 1), of periodic points with minimal period 4 given by V = (x 1 , x 2 , x 3 , x 4 ) ∈ R 4 + : Proof. The proof follows the same lines of the proof of the previous theorem by considering the periodic points of the restrictionφ k of ϕ k to S k (1,1) and the restrictions ϕ k of ϕ (4) k to each S k (p,q) with (p, q) = (1, 1). By Proposition 2 the restrictionφ k is given in the coordinates (x 1 , x 2 ) by (13) and ϕ k is given in the coordinates (x 1 , x 4 ) by (14).
By Proposition 4-3-(i),φ k has a unique fixed point (x 1 , x 2 ) = (2  The full set of these 4-periodic points is a 2-dimensional set V ⊂ S k (p,q) . Like in the proof of the previous theorem, the explicit form of V is easily obtained from