Deformations of cluster mutations and invariant presymplectic forms

We consider deformations of sequences of cluster mutations in finite type cluster algebras, which destroy the Laurent property but preserve the presymplectic structure defined by the exchange matrix. The simplest example is the Lyness 5-cycle, arising from the cluster algebra of type A2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_2$$\end{document}: this deforms to the Lyness family of integrable symplectic maps in the plane. For types A3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_3$$\end{document} and A4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_4$$\end{document}, we find suitable conditions such that the deformation produces a two-parameter family of Liouville integrable maps (in dimensions two and four, respectively). We also perform Laurentification for these maps, by lifting them to a higher-dimensional space of tau functions with a cluster algebra structure, where the Laurent property is restored. More general types of deformed mutations associated with affine Dynkin quivers are shown to correspond to four-dimensional symplectic maps arising as reductions in the discrete sine-Gordon equation.


Lyness maps and Zamolodchikov periodicity
It was observed by Lyness in 1942 [28] that the recurrence generates the sequence which repeats with period five.The Lyness 5-cycle also arises in Coxeter's frieze patterns [3], or as a simple example of Zamolodchikov periodicity in integrable quantum field theories [37], which can be understood in terms of the associahedron K 4 and the cluster algebra defined by the A 2 Dynkin quiver [10], and this leads to a connection with Abel's pentagon identity for the dilogarithm [29].The birational map of the plane corresponding to the recurrence (1.1), that is also appears in the theory of the Cremona group: as conjectured by Usnich and proved by Blanc [1], the birational transformations of the plane that preserve the symplectic form ω = 1 xy dx ∧ dy, (1.4) are generated by SL(2, Z), the torus and transformation (1.3).More generally, the birational map with two parameters a, b is also referred to as the Lyness map.By rescaling (x, y) → (ax, ay), the parameter a = 0 can be removed, so that this is really a one-parameter family, which is described in [6] as "the simplest singular map of the plane."There are also analogous recurrences in higher dimensions, given by the family x n+j + b, which have been shown to admit N 2 independent first integrals for each order N [34].Unlike the special case b = a 2 , which can be rescaled to (1.3), in general the orbits of (1.5) do not all have the same period, and generic orbits are not periodic over an infinite field (e.g.Q, R or C).Moreover, while the iterates in (1.2) are Laurent polynomials in the initial values x 0 , x 1 with integer coefficients, which is one of the characteristic features of the cluster variables in a cluster algebra, the iterates of (1.5) are not Laurent polynomials unless b = a 2 .However, the general map does preserve the same symplectic form (1.4), and there is a conserved quantity K = K(x, y) given by K = xy(x + y) + a(x 2 + y 2 ) + (a 2 + b)(x + y) + ab xy . (1.6) Thus the Lyness map (1.5) is integrable in the Liouville sense, and can be considered as a deformation of the periodic map (1.3) which arises from mutations in a finite type cluster algebra.The purpose of this work is to consider how other integrable maps can be obtained from deformations of cluster mutations.The Zamolodchikov periodicity of Y-systems or T-systems associated with finite type root systems has been extended and generalized in various ways (see [14,26,30] and references), but as far as we are aware the deformations we consider are new.
Following the framework of cluster algebras, we start from a quiver Q (without 1-or 2-cycles) associated with a skew-symmetric exchange matrix B = (b ij ) ∈ Mat N (Z) and an N -tuple of cluster variables x = (x 1 , x 2 , . . ., x N ).Here we consider the cluster variables x i taking values in a field F; the main cases of interest are F = R or C, but for some of our later analysis it will be convenient to consider x i ∈ Q ⊂ Q p .The initial seed is denoted (B, x).Now, for each integer k ∈ [1, N ] we define a mutation µ k which produces a new seed (B , x ) = µ k (B, x), where B = (b ij ) with and x = (x j ) with Here, [a] + = max(a, 0), f k : F × F → F is a differentiable function and k , the first relation in (1.8) becomes the usual exchange relation x k x k = M + k + M − k for cluster mutations in a coefficient-free cluster algebra.In this case, we know that there is a log-canonical presymplectic form compatible with cluster mutations [9,15,22].We extend this result to include more general types of mutations.
Lemma 1.1.Let Q be a quiver associated with the exchange matrix B = (b ij ) and (B , x ) = µ k (B, x), as defined by (1.7) and (1.8).Then if and only if for an arbitrary differentiable function g k : F → F.
Remark 1.2.Equivalently, the function f k can be written in the form for gk arbitrary.
Proof: Using to denote a sum over indices with index k omitted, we have and similarly Hence if we consider the sets k or vice versa, and defining we have of which the general solution is given by (1.10) with g k arbitrary.
According to Lemma 1.1, if the exchange matrix B remains invariant under a sequence of mutations of the form (1.10) then the map that is generated by the same sequence of cluster mutations will preserve a presymplectic form, i.e. the following theorem holds.
Theorem 1.3.Let µ i 1 , µ i 2 , . . ., µ i , for i j ∈ {1, . . ., N }, j ∈ N, be a sequence of mutations defined from (1.7) and (1.8), with each function f i j being of the form (1.10), such that Then the map ϕ : x → x preserves the two-form Remark 1.4.The preceding result admits a slight generalization to the case of cluster algebras (or quivers Q) with periodicity under mutations.In the most general setting, as described by Nakanishi [29], these are defined by an exchange matrix with the property that , where ρ is some permutation of (1, 2, . . ., N ) acting on the indices (equivalently, on the nodes of the quiver Q).The particular case for the cyclic permutation ρ : (1, 2, . . ., N ) → (N, 1, 2, . . ., N − 1) was called cluster mutation-periodicity with period m by Fordy and Marsh [13], who gave a complete classification of the case m = 1.A straightforward adaptation of the above argument shows that if B is periodic, then the map ϕ = ρ−1 µ i • • • µ i 2 µ i 1 leaves B invariant and preserves the corresponding log-canonical presymplectic form (1.11), in the sense that ϕ * (ω) = ω.Lemma 2.3 in [12] covers the special case of this result for ordinary cluster mutations when B is cluster mutation-periodic with period 1, so ϕ = ρ −1 µ 1 and the map can be written as a single recurrence relation.We shall consider an example of this with a generalized mutation in section 3. The slightly different (but closely related) problem of when an ordinary difference equation preserves a log-canonical Poisson bracket was considered in [7].
In the next section our aim is to generalize the example of the Lyness map (1.5), corresponding to the root system A 2 , to other finite type root systems of type A, by taking mutations defined by affine functions f k with additional parameters that destroy the Laurent property but preserve the two-form (1.11).Section 3 contains more general choices of mutations, starting from affine Dynkin diagrams, where the factors g k in (1.10) involve Möbius transformations, which lead to travelling wave reductions of the discrete sine-Gordon equation.We end with a few final remarks.

Deformations of type A periodic maps
In this section, extra parameters are included in the regular exchange relation by taking Hence, according to Theorem 1.3, quivers which are periodic under a particular sequence of mutations (or more generally, are periodic up to a permutation) give rise to parametric cluster maps that preserve the presymplectic form (1.11).If the corresponding exchange matrix is non-singular the parametric cluster maps are symplectic.We begin by examining the case of A 2 in more detail, and then apply this approach to study the integrability of parametric cluster maps associated with the A 3 and A 4 quivers.

Deformed mutations for A 2 quiver
The exchange matrix of type A 2 is In this case, B corresponds to a cluster mutation-periodic quiver with period 1 and So, by the modification of Theorem 1.3 as in Remark 1.4, taking ρ : (1, 2) → (2, 1), for any differentiable function g : F → F the map ϕ = ρ −1 µ 1 given by is symplectic with respect to ω = 1 x 1 x 2 dx 1 ∧ dx 2 .(Compared with (1.10) we have f 1 (x, 1) = xg 1 (1/x) = g(x): in general, replacing g k (x) → xg k (1/x) corresponds to sending B → −B, which is equivalent to replacing the corresponding quiver Q → Q opp , the same quiver with all arrows reversed; see also Remark 1.2.)With (x, y) = (x 1 , x 2 ) and g(x) = ax + b, we reproduce the Lyness map (1.5).Starting from the periodic map (1.3), and relabelling the initial data as (x 0 , x 1 ), any cyclic symmetric function of the iterates x 0 , x 1 , x 2 , x 3 , x 4 in the periodic orbit (1.2) gives a first integral.So in the periodic case there are two independent integrals, namely Both of the latter are sums of Laurent monomials, so in the case of the map with parameters, first integrals can be sought by taking arbitrary linear combinations of the same monomials and solving the resulting conditions on the coefficients.Thus in the case of (1.5), the first integral (1.6) can be considered as a deformation of K 1 above; but a first integral composed of the Laurent monomials in K 2 only exists when b = a 2 and the map is periodic, corresponding to the undeformed situation.
Although the Laurent phenomenon does not persist for the iterates of the Lyness recurrence when b = a 2 , it was pointed out in [12] that there is a connection to a cluster algebra via a lift to a space of higher dimension, defined by the substitution which leads to the Somos-7 recurrence As explained in [13], Somos-type recurrences such as the above, with a sum of two monomials on the right-hand side, can be generated by mutations in a cluster algebra.In the case of (2.4), it is a cluster algebra of rank 7, extended by the addition of the parameters a, b as frozen variables.The rest of this section is devoted to the analogous constructions for A 3 and A 4 .

A 3 quiver with parameters
For the A 3 quiver with exchange matrix where the composition ϕ = µ 3 µ 2 µ 1 acts on the cluster variables (2.5) Since ϕ(B) = B, so the exchange matrix B remains invariant under this sequence of mutations, by Theorem 1.3 the map ϕ preserves the corresponding log-canonical two-form, that is where The original coefficient-free cluster algebra is given by setting a i = 1 = b i for i = 1, 2, 3, and in that case the map ϕ is periodic with period 6, that is ϕ 6 (x) = x.Moreover, one can write down three independent first integrals for the periodic map, by taking appropriate symmetric functions along each orbit, such as 5 i=0 (ϕ * ) i (x j ), 5 i=0 (ϕ * ) i (x j ), etc.However, before considering the deformed case (2.5), there are two ways to simplify the calculations.First of all, assuming the case of generic parameter values a i b i = 0 for all i, we apply the scaling action of the three-dimensional algebraic torus (F * ) 3 , given by x i → λ i x i , λ i = 0, and use this to remove three parameters, so that we obtain where c, d, e are arbitrary.Having simplified the space of parameters, the map ϕ is equivalent to iteration of the system of recurrences Secondly, because we are in an odd-dimensional situation where B necessarily has determinant zero, so that ω is degenerate, so following [12] (cf.Theorem 2.6 therein) we can use ker to generate the one-parameter scaling group (x 1 , x 2 , x 3 ) → (λx 1 , x 2 , λx 3 ) and the projection π onto its monomial invariants, On the y, w-plane, ϕ induces the reduced map φ : which is symplectic, preserving the nondegenerate two-form In the original case where all parameters are 1, the reduced map (2.7) with c = d = e = 1 has period 3, because x 2,n+3 = x 2,n and x 3,n+3 /x 1,n+3 = x 3,n /x 1,n for all n.Thus in that case there are two functionally independent first integrals in the plane, which can be taken as (2.9) (while the product 2 i=0 ( φ * ) i (w) = 1, so does not give a nontrivial integral).Next, we modify K 1 and K 2 by inserting constant coefficients in front of each of their terms, which are all Laurent monomials in K 1 , while for K 2 we can replace the term w + 1 in the denominator by an arbitrary linear function of w.If we require that (at least) one of these modified integrals should be preserved by the map φ, then this puts a finite number of constraints on the coefficients and parameters c, d, e, which are necessary and sufficient for the deformed symplectic map to be Liouville integrable.Thus we obtain the following result.
Theorem 2.1.The condition c = e is necessary and sufficient for the symplectic map (2.7) to admit a deformation of the first integral K 1 , given by hence φ is integrable whenever this condition holds.Requiring that a deformation of K 2 should be preserved imposes the stronger conditions in which case both and K 1 given by (2.10) with c = d 2 are preserved, and all the orbits of φ are periodic with period 3.
Proof: Starting from a general sum of monomials (where we have fixed the scale by assuming that the first term has coefficient 1, and there is the freedom to add an arbitrary constant), we apply the map (2.7) and require that φ * (K 1 ) = K 1 .Comparing the rational functions one each side of the latter equation imposes the requirement c = e and fixes α = β = d, γ = c+d 2 , δ = d, = cd; then choosing to add the constant c + 1 means that K 1 can be factored as in (2.10).Applying the same approach to K 2 requires the additional constraint c = d 2 , restricting to the one-parameter family of period preserves the pencil of biquadratic curves defined by (2.10), which means that there is a map of QRT type [5,31] preserving the same pencil, given by the composition of the horizontal and vertical switch on each curve in the pencil, namely ψ : From general considerations about automorphisms of elliptic curves, since they each correspond to translation by a point, these two maps should commute with one another, and indeed it is straightforward to verify that However, it appears that generically the two maps correspond to translation by two independent points of infinite order, so (over Q, say) this should generate a family of curves with Mordell-Weil group of rank at least 2. (As a special case, when c = d = 1 the map ψ has period 2 for any initial data, corresponding to translation by a 2-torsion point, whereas the period 3 map φ corresponds to addition of a 3-torsion point; so the points are independent, albeit not of infinite order in this case.) We now treat the singularity pattern of the iterates of (2.12), in order to obtain its Laurentification in the sense of [17], i.e. a lift to a map with the Laurent property in a space of higher dimension, in which the new variables can be regarded as tau functions.Rather than a standard singularity confinement analysis, we study orbits defined over Q, and consider a p-adic analogue of confinement, as in [24].The possible singularity patterns can then be obtained using the empirical approach introduced in [20], simply by inspecting the prime factorization of a few terms along a particular orbit.
Thus we choose some particular values for the coefficients and initial data: taking c = 2, d = 3 and (y 0 , w 0 ) = (1, 1), we find the first few iterates are (8,5) so that the first type of p-adic singularity corresponds to τ n ≡ 0 mod p for some n, and the second occurs when σ n ≡ 0 mod p.Our next goal is to show that the tau functions in (2.14) satisfy a system of bilinear equations, namely (we expect that these could be viewed as a reduction of coupled discrete Hirota equations [4,36]), and to prove that this system has the Laurent property.The first equation in (2.15) is straightforward to obtain, as it arises directly from substituting the tau function expressions (2.14) into the second component of (2.12), rewritten in the form of a recurrence, but the second bilinear equation requires more work.If we look at the singularity pattern in the original three-dimensional system (2.6) with e = c, then we see that with a new prefactor ρ n appearing, while x 2,n = y n is already accounted for.Substituting in these formulae to rewrite the system (2.6) in terms of ρ n , σ n , τ n yields (2.16)For the above system, the initial values are ρ 0 , σ 0 , σ 1 , τ −2 , τ −1 , τ 0 , τ 1 , and in principle one could use this to give a direct proof that the sequences (σ n ) and (τ n ) are Laurent polynomials in the initial data, although the sequence ρ n is not.However, note that the product     where B * = (b * ij ) is the skew-symmetric matrix

.18)
The quiver corresponding to this matrix is shown in Figure 2. It is not hard to see that, when c = 1 = d, the bilinear equations (2.15) for n = 0 are generated by applying a mutation at node 1, denoted by μ1 (to distinguish it from mutations in the original A 3 quiver), followed by mutation μ5 : see Figure 3.To prove the Laurent property for the case of arbitrary coefficients, it is necessary to extend the quiver with two extra frozen nodes.
Theorem 2.3.The sequences of tau functions (σ n ) and (τ n ) for the integrable map (2.12) consist of elements of the Laurent polynomial ring , being generated by a sequence of mutations in a cluster algebra defined by the quiver in Figure 2 with the addition of two frozen nodes.
Proof: In order to include the coefficients, we define an extended cluster x = (x 1 , . . ., x8 ) = (τ −2 , . . ., τ 1 , σ 0 , σ 1 , c, d), where x7 = c and x8 = d are frozen variables, and take an extended exchange matrix where two more rows have been appended to (2.18).(The diagram of the quiver with the additional arrows to/from the frozen nodes does not look quite so clear compared with Figure 2, so it has been omitted.)Applying the mutation μ1 gives the exchange relation and produces a new cluster (σ 2 , τ −1 , τ 0 , τ 1 , σ 0 , σ 1 , c, d) and a new matrix μ1 ( B * ) corresponding to the quiver in Figure 3(a) with appropriate arrows to/from the frozen nodes 7 and 8. Next, by applying the mutation μ5 , the exchange relation is with the new cluster being (σ 2 , τ −1 , τ 0 , τ 1 , τ 2 , σ 1 , c, d), and the new exchange matrix μ5 μ1 ( B * ) corresponding to the quiver in Figure 3(b) with suitable extra arrows added to take the coefficients into account.Continuing in a similar way, we find a sequence of mutations to successively generate σ 3 , τ 3 , σ 4 , τ 4 , and so on, such that overall after applying the composition of 12 mutations given by μ463524136251 := μ4 μ6 μ3 μ5 μ2 μ4 μ1 μ3 μ6 μ2 μ5 μ1 ( (in order from right to left), the quiver returns to its starting position; so we have with the index of each of the tau functions increased by 6. Hence by induction both sequences (σ n ), (τ n ) are generated by repeatedly applying this composition of mutations, and the Laurent property follows from the fact that the tau functions are all elements of the cluster algebra, for which it is also known that the Laurent polynomials in the initial data have positive integer coefficients [16,27].
Remark 2.4.Preliminary calculations suggest that the iterates of the QRT map (2.13), which commutes with φ, have a different singularity structure, corresponding to a tau function substitution of the form where η n has weight two.It would be interesting to see whether this has a cluster algebra interpretation.

A 4 quiver with parameters
For the exchange matrix corresponding to the quiver of type A 4 , once again we start from functions of the form with arbitrary coefficients such that a k b k = 0.By rescaling x j → λ j x j with λ j ∈ F * , we can set four of the parameters to 1, so that it is sufficient to consider a four-parameter family of mutations, given by (2.21) Then, defining the action of ϕ = µ 4 µ 3 µ 2 µ 1 on the cluster x = (x 1 , x 2 , x 3 , x 4 ) as above, so the nondegenerate exchange matrix B remains invariant under this sequence of mutations, and according to Theorem 1.3 the map is symplectic with respect to Equivalently, by computing the inverse matrix P = B −1 = (p ij ), the map ϕ preserves the nondegenerate Poisson bracket given by { x i , x j } = p ij x i x j , which has the explicit form with all other brackets being zero.
In the original case of the undeformed quiver, corresponding to a 1 = a 4 = b 1 = b 4 = 1 in (2.21), the map ϕ is completely periodic with period 7, and admits four independent integrals in dimension four.Here we focus on since in the undeformed case these Poisson commute with respect to the bracket (2.23), that is Being a sum/product of cluster variables in the (finite) A 4 cluster algebra, both of these integrals are Laurent polynomials in terms of the initial cluster x, so to deform them we can just take arbitrary linear combinations of the Laurent monomials that appear.
Theorem 2.5.The conditions on the parameters a i , b i (for i = 1, 4) in (2.21) are necessary and sufficient for the first integrals defined by (2.24) in the periodic case to deform to a pair of rational conserved quantities for the symplectic map ϕ = µ 4 µ 3 µ 2 µ 1 that are in involution, i.e. they satisfy (2.25) with respect to the Poisson bracket (2.23).Hence the resulting two-parameter family of maps ϕ is Liouville integrable, with the two functionally independent commuting integrals Proof: The calculation of the conditions on the coefficients of the monomials appearing in the deformed versions of the integrals (2.24) is direct, and leads to the above forms of I 1 , I 2 together with the requirement that b 1 and b 4 should both equal 1.An explicit calculation of their Poisson bracket then shows that the deformed integrals are also in involution, as required for Liouville integrability.To determine the singularity structure of the integrable map ϕ we consider a particular rational orbit with parameters a 1 = 2, a 4 = 3 and all initial x j equal to 1 (see Table 1).Applying the empirical p-adic method from [20] once more, we observe that in the numerators of x 2 and x 3 there are certain primes that do not appear elsewhere, e.g.there are isolated values of n where |x 2,n | p = p −1 for p = 29, 643, 5233, 61613, and similarly there are isolated n where |x 3,n | p = p −1 for p = 17, 71, 79, 89, 3529, 1431173.On the other hand, for p = 61, 151, 251, 571 there are particular values of n where Also for p = 137, 353, 7507 there is a pattern where p first appears in the numerator of x 4 , then in its denominator at the next step, then successively in the denominators of x 3 , x 2 , x 1 , before appearing in the numerator of x 1 , then disappearing at the 7th step (some of the factorizations required to see this are omitted from Table 1 for reasons of space); the product of primes 19 • 23 exhibits the same pattern, although these primes also appear separately elsewhere.These four singularity patterns in the iterates of ϕ suggest introducing four tau functions η n , θ n , σ n , τ n , where the first two have weight two and the last two have weight one, such that π : and direct substitution into the recurrence versions of (2.21) with b 1 = 1 = b 4 , replacing x j → x j,n , x j → x j,n+1 , gives the system (2.28) Initial evidence that this system has the Laurent property is provided by setting corresponding to all initial x j,0 = 1, j = 1, 2, 3, 4 as in Table 1, and iterating the above with a 1 = 2, a 4 = 3, which produces integer-valued tau functions as in Table 2.If the initial data for (2.28) is regarded as a cluster, that is where B * = (b * ij ) is the exchange matrix (since the matrix is skew-symmetric, for brevity we put an asterisk to represent the terms below the diagonal).As in the A 3 case, this is sufficient to generate a sequence of mutations for the tau functions in the original undeformed system, but in order to include the parameters a 1 , a 4 it it necessary to add these as frozen variables.
Theorem 2.6.The sequences of tau functions (τ n ), (η n ), (θ n ), (σ n ) for the integrable map ϕ = µ 4 µ 3 µ 2 µ 1 defined by (2.21) with b 1 = b 4 = 1 consist of elements of the Laurent polynomial ring , being generated by a sequence of mutations in a cluster algebra defined by the exchange matrix (2.29) with the addition of two frozen variables, corresponding to the quiver shown in Figure 4.

Proof:
We take an extended cluster x = (x 1 , . . ., x13 ) = (σ 0 , . . ., σ 5 , η 0 , θ 0 , τ −1 , τ 0 , τ 1 , a 1 , a 4 ), with the coefficients a 1 , a 4 corresponding to additional frozen nodes in the quiver associated with B * = (b * ij ), the extended exchange matrix given by (here we have shown the full matrix so that the exponents of all the exchange relations are visible in each column).The initial quiver is shown in Figure 4. Mutating at node 1 gives the exchange relation μ1 : producing the new cluster μ1 (x) = (τ 2 , σ 1 , . . ., σ 5 , η 0 , θ 0 , τ −1 , τ 0 , τ 1 , a 1 , a 4 ), and subsequently applying mutations μ7 , μ8 , μ9 successively generates exchange relations corresponding to the other three equations in (2.28) for n = 0, with the result being the cluster μ9 μ8 μ7 μ1 (x) = (τ 2 , σ 1 , . . ., σ 5 , η 1 , θ 1 , σ 6 , τ 0 , τ 1 , a 1 , a 4 ).To generate each new instance of the four equations in (2.28) with the index n increased by 1, it is necessary to apply a similar block of four mutations.Let us define the following composition of four mutations by μij := μi μ8 μ7 μj , and to index mutations we use 10, 11 to distinguish nodes 10 and 11 from nodes with single-digit labels.Then if we take a particular composition of 36 mutations given by 9 of these blocks of four, namely μ := μ611 μ510 μ49 μ36 μ25 μ14 μ113 μ102 μ91 = μ6871158710487938762875187411873108729871 (where in the second expression the notation from (2.20) has been reused), then the quiver returns to its starting position; so we have μ( B * ) = B * , μ(x) = (σ 9 , σ 10 , σ 11 , σ 12 , σ 13 , σ 14 , η 9 , θ 9 , τ 8 , τ 9 , τ 10 , a 1 , a 4 ), with the index of each of the tau functions increased by 9. Thus by repeatedly applying these 9 blocks of four mutations, all of the tau functions for the integrable map are produced from clusters in the cluster algebra defined by (2.30).3 Reductions of the discrete sine-Gordon equation In this section we consider two examples of four-dimensional maps that arise as reductions of the lattice sine-Gordon equation introduced in [18], that is where a j , j = 1, 2, 3 are arbitrary parameters.Travelling waves of (3.1) are obtained by imposing periodicity under shifts by N steps in one lattice direction together with M steps in the other direction, so that this is called the (N, M ) reduction.
The two examples we consider below each correspond to particular orientations of the affine A (1) 3 Dynkin diagram, as in Figure 5 (where the notation Ãp,q means there are p clockwise arrows and q anticlockwise arrows).

(2,2) periodic reduction of the lattice sine-Gordon equation
Let us consider the quiver with exchange matrix this is mutation equivalent to Ã2,2 as in Figure 5(a), which corresponds to the exchange matrix µ 3 (B).Then for k = 1, 2, 3, 4 we take the function for arbitrary parameters a 1 , a 2 , a 3 , so that the exchange relation (1.8) contains the function Next, we consider a sequence of mutations which leaves matrix B invariant, specifically So, according to Theorem 1.3, the map ϕ : x → x preserves the two form In this case, the map ϕ corresponds to the (2, 2) periodic reduction of the lattice sine-Gordon equation (3.1) (see Figure 6).
x 1 The matrix B (and hence ω) is degenerate, of rank two.To obtain a symplectic map, we take a pair of monomials corresponding to an integer basis for im B =< (1, 0, 1, 0) T , (0, 1, 0, 1) T >, namely π : Under the projection π defined above, ω is the pullback of the symplectic form which is preserved by the induced map φ : The above map has the first integral so it is Liouville integrable.In fact it is of QRT type: the level sets K = const are symmetric biquadratic curves, and φ where the involutions ι h , ι v correspond to the horizontal and vertical switches on each level set, and ι : y 1 ↔ y 2 .For Laurentification of symmetric QRT maps, see [17].
In four dimensions, the other degrees of freedom in the original map ϕ have essentially trivial dynamics, since
Closed-form expressions for integrals of periodic reductions of the sine-Gordon equation were presented in [23] and their involutivity was proved in [35].

Concluding remarks
We have considered autonomous recurrences or maps obtained by including additional constant parameters in sequences of cluster mutations that generate completely periodic dynamics, and have shown that it is possible to preserve the presymplectic structure defined by the exchange matrix, and also (by imposing suitable constraints on the parameters) obtain Liouville integrable maps.Our starting point for showing Liouville integrability has been the fact that the original periodic maps admit first integrals defined by cyclic symmetric functions of variables along a period of the orbit.Only the examples of A 2 , A 3 and A 4 have been dealt with here, but it would be instructive to make a more systematic study of such functions from the viewpoint of the associated Poisson algebra in order to extend these results to cluster algebras defined by other finite type Dynkin diagrams.We have also treated more general types of mutations, involving Möbius transformations, and shown that for some particular affine type exchange matrices these lead to reductions of the discrete sine-Gordon equation.The parameters a k , b k appearing in our deformed mutations have been assumed constant, but Theorem 1.3 applies equally well to non-autonomous recurrences/maps.In particular, taking 1) leads to the expression for a mutation µ k in a cluster algebra with coefficients [11], which themselves mutate according to The dynamics of the coefficients generates the associated Y-system [26].In [21], it is shown that non-autonomous dynamics also arises from autonomous Y-systems in the case where the exchange matrix is degenerate: one of the simplest examples is provided by the Y-system y n+7 y n = (1 + y n+6 )(1 + y n+1 ) (1 + y −1 n+4 )(1 + y −1 n+3 ) corresponding to the Somos-7 recurrence (2.4), solved in terms of the q-Painlevé V equation which is a non-autonomous version of the Lyness recurrence.The fact that the period of α n is 6 is important, since if q = 1 and α n is periodic with a period that is not a divisor of 6, then (4.1) appears to exhibit chaotic dynamics [2].
As another example based on the A 2 exchange matrix, taking g 1 (x) = ax+b cx+d and letting the coefficients a, b, c, d depend on the index n gives the sequence of symplectic maps ϕ n (x, y) = y, a n y + b n x(c n y + d n ) that corresponds to the non-autonomous nonlinear recurrence .
Invariants of this recurrence when the coefficients are periodic were presented in [8] and have also been studied in the framework of QRT (and non-QRT) maps with periodic coefficients [32,33].

Figure 2 :
Figure 2: The initial quiver Q associated with the exchange matrix (2.18). )

Figure 3 :
Figure 3: The effect of two mutations on the quiver corresponding to (2.18).

Figure 4 :
Figure 4: The initial quiver associated with the extended exchange matrix (2.30).

Figure 6 :
Figure 6: The (2, 2) staircase periodic reduction of the quadrilateral equation (3.1) and so on.For the primes p = 113, 137, 1607, 4001, the values of the p-adic norm |y n | p follow the pattern 1, p −1 , p, p, p −1 , 1, with the corresponding values of |w n | p being 1, 1, p, p −1 , 1, 1, while for the primes p = 2 and 5 there are instances of the same patterns but with p → p 3 and p → p 2 , respectively.(For some of these primes, the whole pattern is not visible above, but it can easily be verified by computing the next few terms, which are omitted here.)In w n there are also other primes that do not appear in y n , e.g.p = 17, 47, 83, 131, 467, 971, and for these the pattern of |w n | p is 1, p −1 , p, 1.This immediately suggests that y n , w n can be written using two different tau functions σ n , τ n , as π :

Table 1 :
Prime factors in an orbit of the integrable deformed A 4 map with a 1 = 2, a 4 = 3.

Table 2 :
Tau functions for the same orbit of the deformed A 4 map as in Table1.