1 Lyness maps and Zamolodchikov periodicity

It was observed by Lyness in 1942 [26] that the recurrence

$$\begin{aligned} x_{n+2}x_n=x_{n+1}+1 \end{aligned}$$
(1.1)

generates the sequence

$$\begin{aligned} x_0,x_1,\frac{x_1+1}{x_0},\frac{x_0+x_1+1}{x_0x_1}, \frac{x_0+1}{x_1}, x_0,x_1,\ldots , \end{aligned}$$
(1.2)

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 [36], 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 [27]. The birational map of the plane corresponding to the recurrence (1.1), that is

$$\begin{aligned} (x,y)\mapsto \left( y, \frac{y+1}{x}\right) , \end{aligned}$$
(1.3)

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

$$\begin{aligned} {\omega }= \frac{1}{xy}\, \textrm{d}x \wedge \textrm{d}y, \end{aligned}$$
(1.4)

are generated by \(SL(2,{{\mathbb {Z}}})\), the torus and transformation (1.3).

More generally, the birational map

$$\begin{aligned} \varphi : \quad (x,y)\mapsto \left( y, \frac{ay+b}{x}\right) , \end{aligned}$$
(1.5)

with two parameters a, b is also referred to as the Lyness map. By rescaling \((x,y)\rightarrow (ax,ay)\), the parameter \(a\ne 0\) 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

$$\begin{aligned} x_{n+N}x_n=\sum _{j=1}^{N-1}x_{n+j} +b, \end{aligned}$$

which have been shown to admit \(\left\lfloor \frac{N}{2}\right\rfloor \) independent first integrals for each order N [32].

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. \({{\mathbb {Q}}},{{\mathbb {R}}}\) or \({\mathbb {C}}\)). 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

$$\begin{aligned} K=\frac{ xy(x+y)+a(x^2+y^2) +(a^2+b)(x+y)+ab}{xy}. \end{aligned}$$
(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, 24, 28] 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 \(B=(b_{ij}) \in \textrm{Mat}_N({\mathbb {Z}})\) and an N-tuple of cluster variables \(\textbf{x} = (x_1,x_2,\ldots , x_N)\). Here, we consider the cluster variables \(x_i\) taking values in a field \({\mathbb {F}}\); the main cases of interest are \({{\mathbb {F}}}={{\mathbb {R}}}\) or \({{\mathbb {C}}}\), but for some of our later analysis, it will be convenient to consider \(x_i\in {{\mathbb {Q}}}\subset {{\mathbb {Q}}}_p\). The initial seed is denoted as \((B, \textbf{x})\). Now, for each integer \(k\in [1,N]\), we define a mutation \({\mu }_k\) which produces a new seed \((B', \textbf{x}')={\mu }_k (B, \textbf{x})\), where \(B'=(b_{ij}')\) with

$$\begin{aligned} b_{ij}' = {\left\{ \begin{array}{ll} -b_{ij} &{}\text {if} \,\,i=k \,\, \text {or} \,\, j=k , \\ b_{ij}+\text {sgn}(b_{ik})[b_{ik}b_{kj}]_+ &{} \text {otherwise}, \end{array}\right. } \end{aligned}$$
(1.7)

and \(\textbf{x}'=(x_j')\) with

$$\begin{aligned} x_j'= {\left\{ \begin{array}{ll} x_k^{-1}\, f_k(M^+_k, M^-_k) &{}\text {for} \,\,j=k \\ \ x_j &{}\text {for} \,\, j \ne k. \end{array}\right. } \end{aligned}$$
(1.8)

Here, \([a]_+ =\max (a,0)\), \(f_k:{\mathbb {F}} \times {\mathbb {F}}\rightarrow {\mathbb {F}}\) is a differentiable function and

$$\begin{aligned} M^+_k:= \prod _{i=1}^N x_i^ {[b_{ki}]_+} \;, \ M^-_k: = \prod _{i=1}^N x_i^ {[-b_{ki}]_+} \;. \end{aligned}$$

For \(f_k(M^+_k,M^-_k)=M^+_k+M^-_k\), 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, 21]. 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 \((B', \textbf{x}')={\mu }_k (B, \textbf{x})\), as defined by (1.7) and (1.8). Then,

$$\begin{aligned} \sum _{i<j} \frac{b'_{ij}}{x'_i x'_j} \textrm{d}x'_i \wedge \textrm{d}x'_j=\sum _{i<j} \frac{b_{ij}}{x_i x_j} \textrm{d}x_i \wedge \textrm{d}x_j \end{aligned}$$
(1.9)

if and only if

$$\begin{aligned} f_k(M^+_k,M^-_k)=M^+_k g_k\left( \frac{M^-_k }{M^+_k }\right) \;, \end{aligned}$$
(1.10)

for an arbitrary differentiable function \(g_k:{\mathbb {F}}\rightarrow {\mathbb {F}}\).

Remark 1.2

Equivalently, the function \(f_k\) can be written in the form

$$\begin{aligned} f_k(M^+_k,M^-_k)=M^-_k \tilde{g}_k\left( \frac{M^+_k }{M^-_k }\right) \;, \end{aligned}$$

for \(\tilde{g}_k\) arbitrary.

Proof

Using \(\sum '\) to denote a sum over indices with index k omitted, we have

$$\begin{aligned} {\omega }&= \sum _{i<j} \frac{b_{ij}}{x_i x_j} \, \textrm{d}x_i \wedge \textrm{d}x_j \\&= \tfrac{1}{2}\left( \hbox { }\ \Sigma _{i,j}' b_{ij}\textrm{d}\log x_i\wedge \textrm{d}\log x_j +\hbox { }\ \Sigma _i' b_{ik} \textrm{d}\log x_i\wedge \textrm{d}\log x_k\right. \\&\quad \left. +\hbox { }\ \Sigma _j' b_{kj}\textrm{d}\log x_k\wedge \textrm{d}\log x_j \right) \\&= \tfrac{1}{2} \hbox {} \Sigma \hbox {}_{i,j}' b_{ij}\textrm{d}\log x_i\wedge \textrm{d}\log x_j + \text {} \Sigma _i' b_{ik} \textrm{d}\log x_i\wedge \textrm{d}\log x_k, \end{aligned}$$

and similarly,

$$\begin{aligned} {\omega }' =&\, \sum _{i<j} \frac{b_{ij}'}{x_i' x_j'} \, \textrm{d}x_i' \wedge \textrm{d}x_j' \\ =&\, \tfrac{1}{2} \hbox {} \Sigma \hbox {}_{i,j}' b_{ij}'\textrm{d}\log x_i'\wedge \textrm{d}\log x_j' + \text {} \Sigma _i' b_{ik}' \textrm{d}\log x_i'\wedge \textrm{d}\log x_k' \\ =&\, \tfrac{1}{2} \hbox { }\ \Sigma _{i,j}' (b_{ij} +\text {sgn}(b_{ik})[b_{ik}b_{kj}]_+)\textrm{d}\log x_i\wedge \textrm{d}\log x_j\\&- \hbox { }\ \Sigma _i' b_{ik} \textrm{d}\log x_i\wedge (-\textrm{d}\log x_k +\textrm{d}\log f_k). \end{aligned}$$

Hence, if we consider the sets

$$\begin{aligned} \beta _k^+=\{i\in \{1,\ldots N \}: b_{ki}>0 \}, \qquad \beta _k^-=\{i\in \{1,\ldots N \}: b_{ki}<0 \}, \end{aligned}$$

then noting that \([b_{ik}b_{kj}]_+=0\) unless either \(i\in \beta _k^+\), \(j\in \beta _k^-\) or vice versa, and defining

$$\begin{aligned} \textrm{d}S^\pm _k:=\pm \textrm{d}\log M_k^\pm =\sum _{i\in \beta _k^\pm }b_{ki}\textrm{d}\log x_i, \end{aligned}$$

we have

$$\begin{aligned} {\omega }'-{\omega }&= \tfrac{1}{2} \hbox { }\ \Sigma _{i,j}' \text {sgn}(b_{ik})[b_{ik}b_{kj}]_+\textrm{d}\log x_i\wedge \textrm{d}\log x_j - \hbox { }\ \Sigma _i' b_{ik} \textrm{d}\log x_i\wedge \textrm{d}\log f_k \\&= \tfrac{1}{2}\left( \sum _{\begin{array}{c} i\in \beta _k^- \\ j\in \beta _k^+ \end{array}}b_{ik}b_{kj}\textrm{d}\log x_i\wedge \textrm{d}\log x_j - \sum _{\begin{array}{c} i\in \beta _k^+ \\ j\in \beta _k^- \end{array}} b_{ik}b_{kj}\textrm{d}\log x_i\wedge \textrm{d}\log x_j \right) \\&\quad +\hbox { }\ \Sigma _i'b_{ki} \textrm{d}\log x_i\wedge \left( \frac{M^+_k}{f_k}\frac{\partial f_k}{\partial M^+_k}\textrm{d}\log M_k^+ + \frac{M^-_k}{f_k}\frac{\partial f_k}{\partial M^-_k}\textrm{d}\log M_k^- \right) \\&= -\sum _{\begin{array}{c} i\in \beta _k^- \\ j\in \beta _k^+ \end{array}}b_{ki}b_{kj}\textrm{d}\log x_i\wedge \textrm{d}\log x_j \\&\quad + (\textrm{d}S^+_k +\textrm{d}S^-_k)\wedge \left( \frac{M^+_k}{f_k}\frac{\partial f_k}{\partial M^+_k}\textrm{d}S_k^+ - \frac{M^-_k}{f_k}\frac{\partial f_k}{\partial M^-_k}\textrm{d}S_k^- \right) \\&= \left( \frac{M^+_k}{f_k}\frac{\partial f_k}{\partial M^+_k} + \frac{M^-_k}{f_k}\frac{\partial f_k}{\partial M^-_k} -1\right) \, \textrm{d}S_k^-\wedge \textrm{d}S_k^+. \end{aligned}$$

Hence, \({\omega }'={\omega }\) if \(f_k=f_k(M_k^+,M_k^-)\) satisfies the linear partial differential equation

$$\begin{aligned} M_k^+ \frac{\partial f_k}{\partial M_k^+ }+M_k^- \frac{\partial f_k}{\partial M_k^- }=f_k, \end{aligned}$$

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 \({\mu }_{i_1}, {\mu }_{i_2}, \ldots ,{\mu }_{i_\ell }\), for \(i_j\in \{1,\ldots ,N\}\), \(j\in {\mathbb {N}}\), 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

$$\begin{aligned} {\mu }_{i_\ell }\ldots {\mu }_{i_2} {\mu }_{i_1}(B,{\textbf{x}})=(B,\tilde{{\textbf{x}}}). \end{aligned}$$

Then, the map \(\varphi :{\textbf{x}}\mapsto \tilde{{\textbf{x}}}\) preserves the two-form

$$\begin{aligned} \omega =\sum _{i<j}^N \frac{b_{ij}}{x_i x_j} \textrm{d}x_i \wedge \textrm{d}x_j. \end{aligned}$$
(1.11)

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 [27], these are defined by an exchange matrix with the property that \({\mu }_{i_\ell }\ldots {\mu }_{i_2} {\mu }_{i_1}(B)=\hat{\rho }(B)\), where \(\hat{\rho }\) is some permutation of \((1,2,\ldots ,N)\) acting on the indices (equivalently, on the nodes of the quiver Q). The particular case \(\mu _m\ldots \mu _2\mu _1 (B)=\rho ^m(B)\), for the cyclic permutation \(\rho : (1,2,\ldots ,N)\mapsto (N,1,2,\ldots , N-1)\) 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 \(\varphi =\hat{\rho }^{-1} {\mu }_{i_\ell }\ldots {\mu }_{i_2} {\mu }_{i_1}\) leaves B invariant and preserves the corresponding log-canonical presymplectic form (1.11), in the sense that \(\varphi ^*({\omega })={\omega }\). 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 \(\varphi =\rho ^{-1}\mu _1\) and the map can be written as a single recurrence relation. We shall consider an example of this with a generalized mutation in Sect. 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 in the discrete sine-Gordon equation. We end with a few final remarks.

2 Deformations of type A periodic maps

In this section, extra parameters are included in the regular exchange relation by taking \(g_k(x)=b_k x+a_k\), since

$$\begin{aligned} f_k(M^+_k,M^-_k)=M^+_k g_k\left( \frac{M^-_k }{M^+_k }\right) =a_k M^+_k+b_kM^-_k\;. \end{aligned}$$
(2.1)

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.

2.1 Deformed mutations for \(A_2\) quiver

The exchange matrix of type \(A_2\) is

$$\begin{aligned} B= \left( \begin{array}{ll} 0 &{}\quad 1 \\ -1 &{} \quad 0 \end{array} \right) . \end{aligned}$$

In this case, B corresponds to a cluster mutation-periodic quiver with period 1 and \(M^+_1=x_2\), \(M^-_1=1\). So, by the modification of Theorem 1.3 as in Remark 1.4, taking \(\rho : (1,2)\mapsto (2,1)\), for any differentiable function \(\tilde{g}:{\mathbb {F}} \rightarrow {\mathbb {F}}\), the map \(\varphi =\rho ^{-1}\mu _1\) given by

$$\begin{aligned} \varphi :(x_1,x_2)\mapsto \left( x_2,\frac{1}{x_1}\tilde{g}(x_2)\right) \; \end{aligned}$$
(2.2)

is symplectic with respect to \(\omega =\frac{1}{x_1 x_2} \textrm{d}x_1 \wedge \textrm{d}x_2\). (Compared with (1.10), we have \(f_1(x,1)=xg_1(1/x)=\tilde{g}(x)\): in general, replacing \(g_k(x)\rightarrow xg_k(1/x)\) corresponds to sending \(B\rightarrow -B\), which is equivalent to replacing the corresponding quiver \(Q\rightarrow Q^\textrm{opp}\), the same quiver with all arrows reversed; see also Remark 1.2.)

With \((x,y)=(x_1,x_2)\) and \(\tilde{g}(x)=ax+b\), 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

$$\begin{aligned} K_1=&\sum _{j=0}^4 x_j=-3+\prod _{j=0}^4 x_j = \frac{x_0^2x_1+x_0x_1^2 +x_0^2+x_1^2+2(x_0+x_1)+1}{x_0x_1}, \\ K_2=&\,\sum _{j=0}^4 x_j x_{j+1} \\ =&\, \frac{x_0x_1(x_0^2x_1^2+x_0^3+x_1^3+x_0^2+x_1^2+x_0+x_1+2) +x_0^3+x_1^3 +2(x_0^2+x_1^2)+x_0+x_1}{x_0^2x_1^2}. \end{aligned}$$

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

$$\begin{aligned} x_{n+2}x_n=ax_{n+1}+b \end{aligned}$$
(2.3)

when \(b\ne a^2\), 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

$$\begin{aligned} x_n = \frac{\tau _{n+5}\tau _n}{\tau _{n+3}\tau _{n+2}}, \end{aligned}$$

which leads to the Somos-7 recurrence

$$\begin{aligned} \tau _{n+7}\tau _n =a\, \tau _{n+6}\tau _{n+1}+b\,\tau _{n+4}\tau _{n+3}. \end{aligned}$$
(2.4)

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 \(A_4\).

2.2 \(A_3\) quiver with parameters

For the \(A_3\) quiver with exchange matrix

$$\begin{aligned} B= \left( \begin{array}{ccc} 0 &{} 1 &{} 0 \\ -1 &{} 0 &{} 1 \\ 0 &{} -1 &{} 0 \\ \end{array} \right) , \end{aligned}$$

as in Fig. 1, we take \(f_k(M^+_k,M^-_k)=a_k M^+_k+b_kM^-_k\). In this case,

$$\begin{aligned} \varphi (B,\textbf{x}):={\mu }_{3}{\mu }_{2}{\mu }_{1}(B,\textbf{x})=\big (B,\varphi (\textbf{x})\big ), \end{aligned}$$

where the composition \(\varphi =\mu _3\mu _2\mu _1\) acts on the cluster variables \(\textbf{x}=(x_1,x_2,x_3)\) according to

$$\begin{aligned} \begin{array}{rcl} \mu _1: \quad (x_1,x_2,x_3)\mapsto (x_1',x_2,x_3), \qquad x_1'x_1 &{} = &{} b_1+a_1 x_2, \\ \mu _2: \quad (x_1',x_2,x_3)\mapsto (x_1',x_2',x_3), \qquad x_2'x_2 &{} = &{} b_2+a_2 x_1'x_3, \\ \mu _3: \quad (x_1',x_2',x_3)\mapsto (x_1',x_2',x_3'), \qquad x_3'x_3 &{} = &{} b_3+a_3 x_2'. \end{array} \end{aligned}$$
(2.5)

Since \(\varphi (B)=B\), so the exchange matrix B remains invariant under this sequence of mutations, by Theorem 1.3, the map \(\varphi \) preserves the corresponding log-canonical two-form, that is

$$\begin{aligned} \varphi ^*({\omega })={\omega }, \end{aligned}$$

where

$$\begin{aligned} \omega =\frac{1}{x_1 x_2} \textrm{d}x_1 \wedge \textrm{d}x_2+\frac{1}{x_2 x_3} \textrm{d}x_2 \wedge \textrm{d}x_3\;. \end{aligned}$$
Fig. 1
figure 1

The \(A_3\) quiver

Full size image

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 \(\varphi \) is periodic with period 6, that is \(\varphi ^6(\textbf{x}) =\textbf{x}\). Moreover, one can write down three independent first integrals for the periodic map, by taking appropriate symmetric functions along each orbit, such as \(\sum _{i=0}^5 (\varphi ^*)^i(x_j)\) and \(\prod _{i=0}^5 (\varphi ^*)^i(x_j)\).

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 \(a_ib_i\ne 0\) for all i, we apply the scaling action of the three-dimensional algebraic torus \(({{\mathbb {F}}}^*)^3\), given by \(x_i\rightarrow {\lambda }_i\, x_i\), \({\lambda }_i\ne 0\), and use this to remove three parameters, so that we obtain

$$\begin{aligned} a_1\rightarrow 1,\quad b_1\rightarrow 1, \quad a_2\rightarrow d, \quad b_2\rightarrow c,\quad a_3\rightarrow 1,\quad b_3\rightarrow e, \end{aligned}$$

where c, d, e are arbitrary. Having simplified the space of parameters, the map \(\varphi \) is equivalent to iteration of the system of recurrences

$$\begin{aligned} \begin{array}{rcl} x_{1,n+1}x_{1,n} &{} = &{} x_{2,n}+1, \\ x_{2,n+1}x_{2,n} &{} = &{} dx_{1,n+1}x_{3,n}+c, \\ x_{3,n+1}x_{3,n} &{} = &{} x_{2,n+1}+e. \end{array} \end{aligned}$$
(2.6)

Secondly, because we are in an odd-dimensional situation where B necessarily has determinant zero, so that \({\omega }\) is degenerate, so following [12] (cf. Theorem 2.6 therein), we can use

$$\begin{aligned} \textrm{ker}\,B=<(1,0,1)^T>, \qquad \textrm{im}\, B= (\textrm{ker}\,B)^\perp = <(0,1,0)^T, (-1,0,1)^T> \end{aligned}$$

to generate the one-parameter scaling group \((x_1,x_2,x_3)\rightarrow ({\lambda }x_1,x_2,{\lambda }x_3)\) and the projection \(\pi \) onto its monomial invariants,

$$\begin{aligned} \pi : \qquad y=x_2, \qquad w=\frac{x_3}{x_1}. \end{aligned}$$

On the y, w-plane, \(\varphi \) induces the reduced map

$$\begin{aligned} \hat{\varphi }: \qquad \left( \begin{array}{c} y\\ w \end{array}\right) \mapsto \left( \begin{array}{c} \big (d(y+1)w+c\big )/y\\ (dw+c)/(y w) +(e-c)/\big (w(y+1)\big )\end{array}\right) , \end{aligned}$$
(2.7)

which is symplectic, preserving the nondegenerate two-form

$$\begin{aligned} \hat{{\omega }}=\textrm{d}\log y \wedge \textrm{d}\log w, \qquad \pi ^*\hat{{\omega }}={\omega }. \end{aligned}$$
(2.8)

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

$$\begin{aligned} \begin{array}{rcccl} K_1&{}=&{}\prod _{i=0}^2 (\hat{\varphi }^*)^i(y)&{}=&{}\frac{(yw+w+1)(y+w+1)}{yw} =-2 +\sum _{i=0}^2 (\hat{\varphi }^*)^i(y), \\ K_2&{}=&{}\sum _{i=0}^2 (\hat{\varphi }^*)^i(w)&{}=&{}\frac{yw^3+yw^2+y^2w+w^2+2w+1}{yw(w+1)} \end{array} \end{aligned}$$
(2.9)

(while the product \(\prod _{i=0}^2 (\hat{\varphi }^*)^i(w)=1\), 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 \(\hat{\varphi }\), 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

$$\begin{aligned} c=e \end{aligned}$$

is necessary and sufficient for the symplectic map (2.7) to admit a deformation of the first integral \(K_1\), given by

$$\begin{aligned} K_1=\frac{(yw+w+d)(y+dw+c)}{yw}, \end{aligned}$$
(2.10)

hence, \(\hat{\varphi }\) is integrable whenever this condition holds. Requiring that a deformation of \(K_2\) should be preserved imposes the stronger conditions

$$\begin{aligned} c=d^2=e, \end{aligned}$$

in which case both

$$\begin{aligned} K_2=\frac{w^3y+d(y+1)w^2+(y^2+2d^2)w+d^3}{yw(w+d)} \end{aligned}$$
(2.11)

and \(K_1\) given by (2.10) with \(c=d^2\) are preserved, and all the orbits of \(\hat{\varphi }\) are periodic with period 3.

Proof

Starting from a general sum of monomials

$$\begin{aligned} K_1= y +{\alpha }\, w +{\beta }\, \frac{w}{y} +\frac{{\gamma }}{y}+\frac{\delta }{w}+\frac{{\epsilon }}{yw}+\textrm{const} \end{aligned}$$

(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 \(\hat{\varphi }^*(K_1)=K_1\). Comparing the rational functions one each side of the latter equation imposes the requirement \(c=e\) and fixes \({\alpha }={\beta }=d\), \({\gamma }=c+d^2\), \(\delta =d\), \({\epsilon }=cd\); 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 3 maps

$$\begin{aligned} \hat{\varphi }: \qquad \left( \begin{array}{c} y\\ w \end{array}\right) \mapsto \left( \begin{array}{c} \big (d(y+1)w+d^2\big )/y\\ d(w+d)/(y w) \big )\end{array}\right) , \end{aligned}$$

which have two independent first integrals given by (2.10) with \(c=d^2\) and (2.11).

Remark 2.2

When \(c=e\), the integrable symplectic map

$$\begin{aligned} \hat{\varphi }: \qquad \left( \begin{array}{c} y\\ w \end{array}\right) \mapsto \left( \begin{array}{c} \big (d(y+1)w+c\big )/y\\ (dw+c)/(y w)\end{array}\right) , \end{aligned}$$
(2.12)

preserves the pencil of biquadratic curves defined by (2.10), which means that there is a map of QRT type [5, 29] preserving the same pencil, given by the composition of the horizontal and vertical switch on each curve in the pencil, namely

$$\begin{aligned} \hat{\psi }: \qquad \left( \begin{array}{c} y\\ w \end{array}\right) \mapsto \left( \begin{array}{c} \bar{y}\\ \bar{w} \end{array}\right) , \qquad \bar{y}y=\frac{(dw+c)(w+d)}{w}, \quad \bar{w}w=\frac{\bar{y}+c}{\bar{y}+1}. \end{aligned}$$
(2.13)

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

$$\begin{aligned} \hat{\psi }\circ \hat{\varphi }=\hat{\varphi }\circ \hat{\psi }. \end{aligned}$$

However, it appears that generically the two maps correspond to translation by two independent points of infinite order, so (over \({{\mathbb {Q}}}\), 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 \(\hat{\psi }\) has period 2 for any initial data, corresponding to translation by a 2-torsion point, whereas the period 3 map \(\hat{\varphi }\) 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 \({{\mathbb {Q}}}\), and consider a p-adic analogue of confinement, as in [22]. The possible singularity patterns can then be obtained using the empirical approach introduced in [19], 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

$$\begin{aligned}{} & {} (8,5),\left( \tfrac{137}{8},\tfrac{17}{40}\right) ,\left( \tfrac{1607}{1096},\tfrac{1048}{2329}\right) , \left( \tfrac{800200}{220159},\tfrac{1068874}{210517}\right) ,\left( \tfrac{3210496223}{160740175},\tfrac{728705399}{780395050}\right) ,\\{} & {} \left( \tfrac{7129742296469}{2344013756975}, \tfrac{2735651842025}{10626437852503}\right) , \end{aligned}$$

so that the values of \(y_n\) for \(n=1,2,3,\ldots \) factorize as

$$\begin{aligned} 2^3, \tfrac{137}{2^3},\tfrac{1607}{2^3\cdot 137}, \tfrac{2^3\cdot 5^2 \cdot 4001}{137\cdot 1607}, \tfrac{11\cdot 17\cdot 113\cdot 137\cdot 1109}{5^2\cdot 1607\cdot 4001}, \tfrac{13\cdot 19\cdot 43\cdot 1607\cdot 417727}{5^2\cdot 11\cdot 17\cdot 113\cdot 1109\cdot 4001}, \ldots , \end{aligned}$$

while the factorizations of the corresponding values of \(w_n\) are

$$\begin{aligned} 5, \tfrac{17}{2^3\cdot 5},\tfrac{2^3\cdot 131}{17\cdot 137}, \tfrac{2\cdot 47\cdot 83\cdot 137}{131\cdot 1607}, \tfrac{467\cdot 971\cdot 1607}{2\cdot 5^2\cdot 47\cdot 83\cdot 4001}, \tfrac{5^2\cdot 4001\cdot 27349681}{11\cdot 17\cdot 113\cdot 467\cdot 971\cdot 1109}, \ldots , \end{aligned}$$

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 \(p\rightarrow p^3\) and \(p\rightarrow p^2\), 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 \({\sigma }_n,\tau _n\), as

$$\begin{aligned} \tilde{\pi }: \qquad y_n=\frac{\tau _{n-2}\tau _{n+1}}{\tau _{n-1}\tau _n}, \quad w_n = \frac{{\sigma }_{n+1}\tau _{n-1}}{{\sigma }_n\tau _{n}}, \end{aligned}$$
(2.14)

so that the first type of p-adic singularity corresponds to \(\tau _n\equiv 0\bmod p\) for some n, and the second occurs when \({\sigma }_n\equiv 0\bmod p\).

Our next goal is to show that the tau functions in (2.14) satisfy a system of bilinear equations, namely

$$\begin{aligned} \begin{array}{rcl} {\sigma }_{n+2}\tau _{n-2}&{} =&{} d\,{\sigma }_{n+1}\tau _{n-1}+c\, {\sigma }_n\tau _n, \\ {\sigma }_n \tau _{n+2} &{} = &{} {\sigma }_{n+2}\tau _n+ d\, {\sigma }_{n+1}\tau _{n+1} \end{array} \end{aligned}$$
(2.15)

(we expect that these could be viewed as a reduction in coupled discrete Hirota equations [4, 35]), 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

$$\begin{aligned} x_{1,n}=\rho _n \, \frac{{\sigma }_{n+1}}{\tau _n}, \qquad x_{3,n}=\rho _n \, \frac{{\sigma }_{n}}{\tau _{n-1}}, \end{aligned}$$

with a new prefactor \(\rho _n\) appearing, while \(x_{2,n}=y_n\) is already accounted for. Substituting in these formulae to rewrite the system (2.6) in terms of \(\rho _n,{\sigma }_n,\tau _n\) yields

$$\begin{aligned} \begin{array}{rcl} \rho _n\rho _{n+1}\, {\sigma }_{n+1}{\sigma }_n&{} =&{} \tau _{n+1}\tau _{n-2}+\tau _n\tau _{n-1}, \\ \tau _{n+2}\tau _{n-2}&{} =&{} \rho _n\rho _{n+1}\, d\, {\sigma }_{n+1}^2 + c\,\tau _n^2, \\ \rho _n\rho _{n+1}\, {\sigma }_{n+2}{\sigma }_{n+1}&{} =&{} \tau _{n+2}\tau _{n-1}+c\,\tau _{n+1}\tau _{n}. \end{array} \end{aligned}$$
(2.16)

For the above system, the initial values are \(\rho _0,{\sigma }_0,{\sigma }_1, \tau _{-2},\tau _{-1},\tau _0,\tau _1\), and in principle, one could use this to give a direct proof that the sequences \(({\sigma }_n)\) and \((\tau _n)\) are Laurent polynomials in the initial data, although the sequence \(\rho _n\) is not. However, note that, the product \(\rho _n\rho _{n+1}\) can be eliminated from any two of the equations in (2.16), so doing this for each pair gives a set of three equations of degree 3, and then eliminating \(\tau _{n+2}\) from any two of the latter results in the first equation in (2.15), while eliminating \(\tau _{n+2}\) instead produces the relation

$$\begin{aligned} {\sigma }_n\tau _{n+2}\tau _{n-2}=d\,{\sigma }_{n+1}(\tau _{n+1}\tau _{n-2}+\tau _n\tau _{n-1})+c\,{\sigma }_n\tau _n^2. \end{aligned}$$

Finally, the second relation in (2.15) follows by combining the first relation with the above to eliminate \(\tau _{n-2}\).

Immediate evidence for the Laurent property can be seen by iterating the system (2.15) for \(c=2\), \(d=3\) with all initial values \(\tau _{-2}=\tau _{-1}=\tau _0 = \tau _1={\sigma }_0={\sigma }_1=1\), corresponding to the initial values \(y_0=w_0=1\) in the orbit considered above. The first few terms are the integers

$$\begin{aligned} \begin{array}{rllllllll} (\tau _n)_{n\ge 1}: \, &{} 1, &{} 8, &{} 137, &{} 1607, &{} 100025, &{} 23434279, &{} 4436678467, &{} 1750170148834, \\ ({\sigma }_n)_{n\ge 1}: &{} 1, &{} 5, &{} 17, &{} 131, &{} 7802, &{} 453457, &{} 27349681, &{} 18332191183, \end{array} \end{aligned}$$

and so on. It is also easy to verify directly that the first few terms \(\tau _2,{\sigma }_1\), etc., obtained by iteration of (2.15) are Laurent polynomials in the initial data with coefficients belonging to \({{\mathbb {Z}}}[c,d]\).

Fig. 2
figure 2

The initial quiver Q associated with the exchange matrix (2.18)

Full size image

To make further progress, it is helpful to consider the initial data for (2.15) as a set of cluster variables \((\tilde{x}_1,\tilde{x}_2,\tilde{x}_3,\tilde{x}_4,\tilde{x}_5,\tilde{x}_6)=(\tau _{-2},\tau _{-1},\tau _0,\tau _1,{\sigma }_0,{\sigma }_1)\) and calculate the pullback of the symplectic form (2.8) by the map \(\tilde{\pi }\) defined by the tau function expressions (2.14), that is

$$\begin{aligned} \tilde{{\omega }}=\tilde{\pi }^*\hat{{\omega }}=\sum _{i<j} b_{ij}^*\textrm{d}\log \tilde{x}_i \wedge \textrm{d}\log \tilde{x}_j, \end{aligned}$$
(2.17)

where \(B^*=(b_{ij}^*)\) is the skew-symmetric matrix

$$\begin{aligned} B^*=\left( \begin{array}{cccccc} 0 &{}\quad 1&{}\quad -1 &{} \quad 0 &{}\quad -1 &{}\quad 1 \\ -1 &{}\quad 0 &{}\quad 2 &{}\quad -1 &{}\quad 1 &{}\quad -1 \\ 1 &{}\quad -2 &{} \quad 0 &{}\quad 1 &{} \quad 1 &{}\quad -1 \\ 0 &{} \quad 1 &{}\quad -1 &{}\quad 0 &{}\quad -1 &{}\quad 1 \\ 1 &{}\quad -1 &{}\quad -1 &{}\quad 1 &{}\quad 0 &{}\quad 0 \\ -1 &{}\quad 1 &{}\quad 1 &{}\quad -1 &{}\quad 0 &{}\quad 0 \end{array} \right) . \end{aligned}$$
(2.18)

The quiver corresponding to this matrix is shown in Fig. 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 \(\tilde{\mu }_1\) (to distinguish it from mutations in the original \(A_3\) quiver), followed by mutation \(\tilde{\mu }_5\): see Fig. 3. To prove the Laurent property for the case of arbitrary coefficients, it is necessary to extend the quiver with two extra frozen nodes.

Fig. 3
figure 3

The effect of two mutations on the quiver corresponding to (2.18)

Full size image

Theorem 2.3

The sequences of tau functions \(({\sigma }_n)\) and \((\tau _n)\) for the integrable map (2.12) consist of elements of the Laurent polynomial ring \({{\mathbb {Z}}}_{>0}[c,d,\tau _{-2}^{\pm 1},\tau _{-1}^{\pm 1},\tau _{0}^{\pm 1},\tau _{1}^{\pm 1},{\sigma }_{0}^{\pm 1},{\sigma }_{1}^{\pm 1}]\), being generated by a sequence of mutations in a cluster algebra defined by the quiver in Fig. 2 with the addition of two frozen nodes.

Proof

In order to include the coefficients, we define an extended cluster \(\tilde{\textbf{x}}=(\tilde{x}_1,\ldots ,\tilde{x}_8)=(\tau _{-2},\ldots ,\tau _1,{\sigma }_0,{\sigma }_1,c,d)\), where \(\tilde{x}_7=c\) and \(\tilde{x}_8=d\) are frozen variables and take an extended exchange matrix

$$\begin{aligned} \tilde{B}^*=\left( \begin{array}{cccccc} 0 &{}\quad 1&{}\quad -1 &{}\quad 0 &{}\quad -1 &{} \quad 1 \\ -1 &{}\quad 0 &{} \quad 2 &{}\quad -1 &{}\quad 1 &{}\quad -1 \\ 1 &{}\quad -2 &{}\quad 0 &{}\quad 1 &{} \quad 1 &{}\quad -1 \\ 0 &{}\quad 1 &{}\quad -1 &{} \quad 0 &{}\quad -1 &{}\quad 1 \\ 1 &{}\quad -1 &{}\quad -1 &{} \quad 1 &{}\quad 0 &{} \quad 0 \\ -1 &{}\quad 1 &{}\quad 1 &{}\quad -1 &{}\quad 0 &{}\quad 0 \\ 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -1 \\ -1 &{}\quad -1 &{} \quad 1 &{} \quad 1 &{} \quad 0 &{}\quad 0 \end{array} \right) , \end{aligned}$$
(2.19)

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 Fig. 2, so it has been omitted.) Applying the mutation \(\tilde{\mu }_1\) gives the exchange relation

$$\begin{aligned} {\sigma }_2\tau _{-2}=d\,{\sigma }_1\tau _{-1}+c\,{\sigma }_0\tau _0, \end{aligned}$$

and produces a new cluster \(({\sigma }_2,\tau _{-1},\tau _0,\tau _1,{\sigma }_0,{\sigma }_1,c,d)\) and a new matrix \(\tilde{\mu }_1(\tilde{B}^*)\) corresponding to the quiver in Fig. 3a with appropriate arrows to/from the frozen nodes 7 and 8. Next, by applying the mutation \(\tilde{\mu }_5\), the exchange relation is

$$\begin{aligned} \tau _2{\sigma }_0=d\, {\sigma }_1\tau _1+{\sigma }_2\tau _0, \end{aligned}$$

with the new cluster being \(({\sigma }_2,\tau _{-1},\tau _0,\tau _1,\tau _2,{\sigma }_1,c,d)\), and the new exchange matrix \(\tilde{\mu }_5\tilde{\mu }_1(\tilde{B}^*)\) corresponding to the quiver in Fig. 3b 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 \({\sigma }_3,\tau _3,{\sigma }_4,\tau _4\), and so on, such that overall after applying the composition of 12 mutations given by

$$\begin{aligned} \tilde{\mu }_{463524136251}:=\tilde{\mu }_4\tilde{\mu }_6\tilde{\mu }_3\tilde{\mu }_5\tilde{\mu }_2\tilde{\mu }_4\tilde{\mu }_1\tilde{\mu }_3\tilde{\mu }_6\tilde{\mu }_2\tilde{\mu }_5\tilde{\mu }_1 \end{aligned}$$
(2.20)

(in order from right to left), the quiver returns to its starting position; so we have

$$\begin{aligned} \tilde{\mu }_{463524136251}(\tilde{B}^*)= \tilde{B}^*, \qquad \tilde{\mu }_{463524136251}(\tilde{\textbf{x}})=(\tau _4,\tau _5,\tau _6,\tau _7,{\sigma }_6,{\sigma }_7,c,d), \end{aligned}$$

with the index of each of the tau functions increased by 6. Hence, by induction, both sequences \(({\sigma }_n)\), \((\tau _n)\) 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, 25].

Remark 2.4

Preliminary calculations suggest that the iterates of the QRT map (2.13), which commutes with \(\hat{\varphi }\), have a different singularity structure, corresponding to a tau function substitution of the form

$$\begin{aligned} y_n=\frac{\eta _{n}}{{\sigma }_n\tau _{n-1}}, \quad w_n = \frac{{\sigma }_{n+1}\tau _{n-1}}{{\sigma }_n\tau _{n}}, \end{aligned}$$

where \(\eta _n\) has weight two. It would be interesting to see whether this has a cluster algebra interpretation.

2.3 \(A_4\) quiver with parameters

For the exchange matrix

$$\begin{aligned} B= \left( \begin{array}{cccc} 0 &{} 1 &{} 0 &{} 0 \\ -1 &{} 0 &{} 1 &{} 0 \\ 0 &{} -1 &{} 0 &{} 1 \\ 0 &{} 0 &{} -1 &{} 0 \end{array} \right) , \end{aligned}$$

corresponding to the quiver of type \(A_4\), once again we start from functions of the form \(f_k(M^+_k,M^-_k)=a_k M^+_k+b_kM^-_k\), with arbitrary coefficients such that \(a_kb_k\ne 0\). By rescaling \(x_j\rightarrow {\lambda }_j \, x_j\) with \({\lambda }_j\in {{\mathbb {F}}}^*\), we can set four of the parameters to 1, so that it is sufficient to consider a four-parameter family of mutations, given by

$$\begin{aligned} \begin{array}{rcl} \mu _1: \quad (x_1,x_2,x_3,x_4)\mapsto (x_1',x_2,x_3,x_4), \qquad x_1'x_1 &{} = &{} b_1+a_1 x_2, \\ \mu _2: \quad (x_1',x_2,x_3,x_4)\mapsto (x_1',x_2',x_3,x_4), \qquad x_2'x_2 &{} = &{} 1+ x_1'x_3, \\ \mu _3: \quad (x_1',x_2',x_3,x_4)\mapsto (x_1',x_2',x_3',x_4), \qquad x_3'x_3 &{} = &{} 1+ x_2'x_4, \\ \mu _4: \quad (x_1',x_2',x_3',x_4)\mapsto (x_1',x_2',x_3',x_4'), \qquad x_4'x_4 &{} = &{} b_4+ a_4 x_3'. \end{array} \end{aligned}$$
(2.21)

Then, defining the action of \(\varphi =\mu _4\mu _3\mu _2\mu _1\) on the cluster \(\textbf{x}=(x_1,x_2,x_3,x_4)\) as above,

$$\begin{aligned} \varphi (B,\textbf{x}):={\mu }_{4}{\mu }_{3}{\mu }_{2} {\mu }_{1}(B,\textbf{x}) =\big (B,\varphi (\textbf{x})\big ), \end{aligned}$$

so the nondegenerate exchange matrix B remains invariant under this sequence of mutations, and according to Theorem 1.3, the map

$$\begin{aligned} \textbf{x} \mapsto \varphi (\textbf{x}) \end{aligned}$$

is symplectic with respect to

$$\begin{aligned} \omega =\frac{1}{x_1 x_2} \textrm{d}x_1 \wedge \textrm{d}x_2+\frac{1}{x_2 x_3} \textrm{d}x_2 \wedge \textrm{d}x_3+\frac{1}{x_3 x_4} \textrm{d}x_3 \wedge \textrm{d}x_4\;. \end{aligned}$$
(2.22)

Equivalently, by computing the inverse matrix \(P=B^{-1}=(p_{ij})\), the map \(\varphi \) preserves the nondegenerate Poisson bracket given by \(\{\,x_i,x_j\,\}=p_{ij}\,x_ix_j\), which has the explicit form

$$\begin{aligned} \{\,x_2,x_1\,\}=x_2x_1, \qquad \{\,x_4,x_1\,\}=x_4x_1, \qquad \{\,x_4,x_3\,\}=x_4x_3, \end{aligned}$$
(2.23)

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 \(\varphi \) is completely periodic with period 7 and admits four independent integrals in dimension four. Here, we focus on

$$\begin{aligned} I_1=\sum _{j=0}^6(\varphi ^*)^j(x_1), \qquad I_2=\prod _{j=0}^6(\varphi ^*)^j(x_1), \end{aligned}$$
(2.24)

since in the undeformed case these Poisson commute with respect to the bracket (2.23), that is

$$\begin{aligned} \{\, I_1,I_2\,\}=0. \end{aligned}$$
(2.25)

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 \(\textbf{x}\), so to deform them, we can just take arbitrary linear combinations of the Laurent monomials that appear.

Theorem 2.5

The conditions

$$\begin{aligned} b_1=1=b_4 \end{aligned}$$
(2.26)

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 \(\varphi =\mu _4\mu _3\mu _2\mu _1\) 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 \(\varphi \) is Liouville integrable, with the two functionally independent commuting integrals

$$\begin{aligned} I_1=&\frac{1}{ {x_1} {x_2} {x_3} {x_4}} \Big ( {a_1} {a_4} {x_1} {x_2}+ {a_1} {a_4}^2 {x_1} {x_2} {x_3}+ {a_1} {x_1} {x_2} {x_3} + {a_1} {a_4} {x_1} {x_2} {x_3}^2+ {a_1} {a_4} {x_1} {x_4} \\&+ {a_1} {a_4} {x_1} {x_2}^2 {x_4} + {a_1} {a_4} {x_3} {x_4}+ {a_1} {a_4} {x_1}^2 {x_3} {x_4}+ {a_4} {x_2} {x_3} {x_4} + {a_1}^2 {a_4} {x_2} {x_3} {x_4}+ {a_4} {x_1}^2 {x_2} {x_3} {x_4} \\&+ {a_1} {a_4} {x_2}^2 {x_3} {x_4} + {a_1} {a_4} {x_1} {x_3}^2 {x_4}+ {a_1} {a_4} {x_1} {x_2} {x_4}^2+ {a_1} {x_1} {x_2} {x_3} {x_4}^2\Big ) , \\ I_2=&\frac{ ( {a_1} + {x_2}) ( {x_1}+ {x_3}) \left( {a_4} + {x_3}\right) \left( {x_2}+ {x_4}\right) \left( {x_1} {x_2}+ {a_4} {x_1} {x_2} {x_3}+ {x_1} {x_4}+ {x_3} {x_4}+ {a_1} {x_2} {x_3} {x_4}\right) }{ {x_1} {x_2}^2 {x_3}^2 {x_4}} . \end{aligned}$$

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.

Table 1 Prime factors in an orbit of the integrable deformed \(A_4\) map with \(a_1=2, a_4=3\)
Full size table

To determine the singularity structure of the integrable map \(\varphi \), 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 [19] 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 \(|x_{1,n}|_p=|x_{2,n}|_p=|x_{3,n}|_p=|x_{4,n}|_p=p\) and also \(|x_{1,n-1}|_p=p^{-1}\), \(|x_{4,n+1}|_p=p^{-1}\). 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 \(19\cdot 23\) exhibits the same pattern, although these primes also appear separately elsewhere. These four singularity patterns in the iterates of \(\varphi \) suggest introducing four tau functions \(\eta _n,\theta _n,{\sigma }_n,\tau _n\), where the first two have weight two and the last two have weight one, such that

$$\begin{aligned} \tilde{\pi }: \quad x_{1,n}=\frac{{\sigma }_n\tau _{n+1}}{{\sigma }_{n+1}\tau _n}, \quad x_{2,n}=\frac{\eta _n}{{\sigma }_{n+2}\tau _n}, \quad x_{3,n}=\frac{\theta _n}{{\sigma }_{n+3}\tau _n}, \quad x_{4,n}=\frac{{\sigma }_{n+5}\tau _{n-1}}{{\sigma }_{n+4}\tau _n}, \nonumber \\ \end{aligned}$$
(2.27)

and direct substitution into the recurrence versions of (2.21) with \(b_1=1=b_4\), replacing \(x_j\rightarrow x_{j,n}\), \(x_j'\rightarrow x_{j,n+1}\), gives the system

$$\begin{aligned} \begin{array}{rcl} \tau _{n+2}{\sigma }_n&{} =&{} \tau _n{\sigma }_{n+2}+a_1\, \eta _n, \\ \eta _{n+1}\eta _n &{}= &{}{\sigma }_{n+1}\tau _{n+2}\theta _n + {\sigma }_{n+2}{\sigma }_{n+3}\tau _n\tau _{n+1}, \\ \theta _{n+1}\theta _n &{} = &{} {\sigma }_{n+5}\tau _{n-1}\eta _{n+1}+{\sigma }_{n+3}{\sigma }_{n+4}\tau _n\tau _{n+1}, \\ {\sigma }_{n+6}\tau _{n-1}&{} = &{} {\sigma }_{n+4}\tau _{n+1}+a_4\, \theta _{n+1}. \end{array} \end{aligned}$$
(2.28)

Initial evidence that this system has the Laurent property is provided by setting \({\sigma }_0=\cdots ={\sigma }_5=\eta _0=\theta _0=\tau _{-1}=\tau _0=\tau _1=1\), 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.

Table 2 Tau functions for the same orbit of the deformed \(A_4\) map as in Table 1
Full size table
Fig. 4
figure 4

The initial quiver associated with the extended exchange matrix (2.30)

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If the initial data for (2.28) is regarded as a cluster, that is

$$\begin{aligned} (\tilde{x}_1,\ldots ,\tilde{x}_{11})=({\sigma }_0,\ldots ,{\sigma }_5,\eta _0,\theta _0,\tau _{-1},\tau _0,\tau _1), \end{aligned}$$

then the pullback of the symplectic form (2.22) under the map \(\tilde{\pi }\) defined by (2.27) is

$$\begin{aligned} \tilde{{\omega }}=\tilde{\pi }^*{\omega }=\sum _{i<j}b_{ij}^*\, \textrm{d}\log \tilde{x}_i\wedge \textrm{d}\log \tilde{x}_j, \end{aligned}$$

where \(B^*=(b^*_{ij})\) is the exchange matrix

$$\begin{aligned} B^*=\left( \begin{array}{ccccccccccc} 0 &{}\quad 0 &{}\quad -1 &{}\quad 0 &{}\quad 0 &{} \quad 0 &{} \quad 1 &{} \quad 0 &{} \quad 0 &{}\quad -1 &{} \quad 0 \\ &{} \quad 0 &{}\quad 1 &{} \quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -1 &{}\quad 0 &{} \quad 0 &{} \quad 1 &{}\quad 0 \\ &{} &{}\quad 0 &{}\quad 1 &{} \quad 0 &{}\quad 0 &{} \quad 0 &{}\quad -1 &{} \quad 0 &{}\quad 0 &{} \quad 1 \\ &{} &{} &{}\quad 0 &{}\quad 1 &{}\quad -1 &{} \quad 1 &{}\quad 0 &{}\quad -1 &{}\quad 0 &{} \quad 0 \\ &{} &{} &{} &{}\quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 1 &{}\quad 0 &{}\quad -1 &{}\quad 0 \\ &{} &{} &{} &{} &{}\quad 0 &{}\quad 0 &{}\quad -1 &{}\quad 0 &{} \quad 1 &{}\quad 0 \\ &{} &{} &{} &{} &{} &{}\quad 0 &{} \quad 1 &{} \quad 0 &{} \quad 0 &{}\quad -1 \\ &{} &{} &{} \quad * &{} &{} &{} &{}\quad 0 &{}\quad 1 &{}\quad 0 &{} \quad 0 \\ &{} &{} &{} &{} &{} &{} &{} &{} \quad 0 &{}\quad 1 &{}\quad 0 \\ &{} &{} &{} &{} &{} &{} &{} &{} &{}\quad 0 &{} \quad 1 \\ &{} &{} &{} &{} &{} &{} &{} &{} &{} &{}\quad 0 \end{array} \right) \end{aligned}$$
(2.29)

(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 is necessary to add these as frozen variables.

Theorem 2.6

The sequences of tau functions \((\tau _n)\), \((\eta _n)\), \((\theta _n)\), \(({\sigma }_n)\) for the integrable map \(\varphi =\mu _4\mu _3\mu _2\mu _1\) defined by (2.21) with \(b_1=b_4=1\) consist of elements of the Laurent polynomial ring \({{\mathbb {Z}}}_{>0}[a_1,a_4, {\sigma }_{0}^{\pm 1},{\sigma }_{1}^{\pm 1},{\sigma }_{2}^{\pm 1},{\sigma }_{3}^{\pm 1},{\sigma }_{4}^{\pm 1},{\sigma }_{5}^{\pm 1}, \eta _{0}^{\pm 1},\theta _{0}^{\pm 1},\tau _{-1}^{\pm 1},\tau _{0}^{\pm 1},\tau _{1}^{\pm 1}]\), 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 Fig. 4.

Proof

We take an extended cluster

$$\begin{aligned} \tilde{\textbf{x}}=(\tilde{x}_1,\ldots ,\tilde{x}_{13})=({\sigma }_0,\ldots ,{\sigma }_5,\eta _0,\theta _0,\tau _{-1},\tau _0,\tau _1,a_1,a_4), \end{aligned}$$

with the coefficients \(a_1,a_4\) corresponding to additional frozen nodes in the quiver associated with \(\tilde{B}^*=(b^*_{ij})\), the extended exchange matrix given by

$$\begin{aligned} \tilde{B}^*=\left( \begin{array}{ccccccccccc} 0 &{}\quad 0 &{}\quad -1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{} \quad 0 &{}\quad -1 &{}\quad 0 \\ 0 &{}\quad 0 &{} \quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -1 &{} \quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 \\ 1 &{}\quad -1 &{} \quad 0 &{}\quad 1 &{}\quad 0 &{} \quad 0 &{} \quad 0 &{}\quad -1 &{}\quad 0 &{}\quad 0 &{}\quad 1 \\ 0 &{} \quad 0 &{}\quad -1 &{}\quad 0 &{}\quad 1 &{}\quad -1 &{} \quad 1 &{} \quad 0 &{}\quad -1 &{} \quad 0 &{}\quad 0 \\ 0 &{} \quad 0 &{}\quad 0 &{} \quad -1&{}\quad 0 &{} \quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad -1 &{}\quad 0 \\ 0 &{}\quad 0 &{} \quad 0 &{}\quad 1 &{}\quad 0 &{} \quad 0 &{}\quad 0 &{}\quad -1 &{}\quad 0 &{} \quad 1 &{}\quad 0 \\ -1 &{} \quad 1 &{}\quad 0 &{}\quad -1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{} \quad 1 &{} \quad 0 &{} \quad 0 &{}\quad -1 \\ 0 &{} \quad 0 &{}\quad 1 &{} \quad 0 &{}\quad -1 &{} \quad 1 &{}\quad -1 &{} \quad 0 &{}\quad 1 &{} \quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{} \quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -1 &{}\quad 0 &{} \quad 1 &{}\quad 0 \\ 1 &{}\quad -1 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad -1 &{}\quad 0 &{}\quad 0 &{}\quad -1 &{}\quad 0 &{} \quad 1 \\ 0 &{}\quad 0 &{} \quad -1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{} \quad -1 &{} \quad 0 \\ -1 &{}\quad -1 &{} \quad 0 &{}\quad 0 &{} \quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{} \quad 0 &{}\quad 1 &{} \quad 1 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{} \quad 1 &{} \quad 1 &{} \quad 0 &{}\quad 0 &{}\quad -1 &{}\quad -1 &{} \quad 0 \end{array} \right) \end{aligned}$$
(2.30)

(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 Fig. 4. Mutating at node 1 gives the exchange relation

$$\begin{aligned} \tilde{\mu }_1:\quad \tau _2{\sigma }_0= \tau _0{\sigma }_{2}+a_1\, \eta _0, \end{aligned}$$

producing the new cluster \(\tilde{\mu }_1(\tilde{\textbf{x}})= (\tau _2,{\sigma }_1,\ldots ,{\sigma }_5,\eta _0,\theta _0,\tau _{-1},\tau _0,\tau _1,a_1,a_4)\), and subsequently applying mutations \(\tilde{\mu }_7,\tilde{\mu }_8,\tilde{\mu }_9\) successively generates exchange relations corresponding to the other three equations in (2.28) for \(n=0\), with the result being the cluster \(\tilde{\mu }_9\tilde{\mu }_8\tilde{\mu }_7\tilde{\mu }_1(\tilde{\textbf{x}})= (\tau _2,{\sigma }_1,\ldots ,{\sigma }_5,\eta _1,\theta _1,{\sigma }_{6},\tau _0,\tau _1,a_1,a_4)\). 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

$$\begin{aligned} \hat{\mu }_{ij} := \tilde{\mu }_i\tilde{\mu }_8\tilde{\mu }_7\tilde{\mu }_j, \end{aligned}$$

and to index mutations, we use \(\overline{10},\overline{11}\) 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

$$\begin{aligned} \hat{\hat{\mu }} := \hat{\mu }_{6\overline{11}}\,\hat{\mu }_{5\overline{10}}\,\hat{\mu }_{49}\, \hat{\mu }_{36}\,\hat{\mu }_{25}\, \hat{\mu }_{14}\,\hat{\mu }_{\overline{11}3}\,\hat{\mu }_{\overline{10}2}\,\hat{\mu }_{91} = \tilde{\mu }_{687\overline{11}587\overline{10}4879387628751874\overline{11}873\overline{10}8729871} \end{aligned}$$

(where in the second expression the notation from (2.20) has been reused), then the quiver returns to its starting position; so we have

$$\begin{aligned} \hat{\hat{\mu }}(\tilde{B}^*)= \tilde{B}^*, \qquad \hat{\hat{\mu }}(\tilde{\textbf{x}})=({\sigma }_9,{\sigma }_{10},{\sigma }_{11},{\sigma }_{12},{\sigma }_{13}, {\sigma }_{14},\eta _9,\theta _9, \tau _8,\tau _9,\tau _{10},a_1,a_4), \end{aligned}$$

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 in the discrete sine-Gordon equation

In this section, we consider two examples of four-dimensional maps that arise as reductions in the lattice sine-Gordon equation introduced in [18], that is

$$\begin{aligned} a_1(x_{n,m} x_{n+1,m+1}-x_{n+1,m}x_{n,m+1})+a_2x_{n,m}x_{n+1,m}x_{n,m+1}x_{n+1,m+1}=a_3\;, \end{aligned}$$
(3.1)

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

$$\begin{aligned} u_{n+N,m+M}=u_{n,m}\implies u_{n,m}=x_k,\quad k=Mn-Nm; \end{aligned}$$

this is called the (N, M) reduction.

The two examples we consider below each correspond to particular orientations of the affine \(A_3^{(1)}\) Dynkin diagram, as in Fig. 5 (where the notation \(\tilde{A}_{p,q}\) means there are p clockwise arrows and q anticlockwise arrows).

Fig. 5
figure 5

Two orientations of the \(A_3^{(1)}\) Dynkin diagram

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3.1 (2, 2) Periodic reduction in the lattice sine-Gordon equation

Let us consider the quiver with exchange matrix

$$\begin{aligned} B= \left( \begin{array}{cccc} 0 &{} 1 &{} 0 &{} 1 \\ -1 &{} 0 &{} -1 &{} 0 \\ 0 &{} 1 &{} 0 &{} 1 \\ -1 &{} 0 &{} -1 &{} 0 \end{array} \right) ; \end{aligned}$$

this is mutation equivalent to \(\tilde{A}_{2,2}\) as in Fig. 5a, which corresponds to the exchange matrix \(\mu _3(B)\). Then, for \(k=1,2,3,4\), we take the function

$$\begin{aligned} g_k(x)=\frac{a_1 x+a_3}{a_2 x+a_1}, \end{aligned}$$

for arbitrary parameters \(a_1,a_2,a_3\), so that the exchange relation (1.8) contains the function

$$\begin{aligned} f_k(M^+_k,M^-_k)=M^+_k g_k\left( \frac{M^-_k }{M^+_k }\right) =M^+_k \frac{a_1 M^-_k +a_3 M^+_k}{a_2 M^-_k+a_1M^+_k }\;. \end{aligned}$$

Next, we consider a sequence of mutations which leaves matrix B invariant, specifically

$$\begin{aligned} \varphi (B,{\textbf{x}}):={\mu }_{3}{\mu }_{1}{\mu }_{4} {\mu }_{2}(B,{\textbf{x}})=(B,\tilde{{\textbf{x}}}), \ \text {where} \ \tilde{{\textbf{x}}}=(\tilde{x}_1,\tilde{x}_2,\tilde{x}_3,\tilde{x}_4) \end{aligned}$$

and

$$\begin{aligned} \tilde{x}_2&=\frac{1}{x_2}\left( \frac{a_1 x_1 x_3+a_3}{a_2 x_1 x_3+a_1} \right) \;, \ \tilde{x}_4=\frac{1}{x_4}\left( \frac{a_1 x_1 x_3+a_3}{a_2 x_1 x_3+a_1} \right) \;, \\ \tilde{x}_1&=\frac{1}{x_1}\left( \frac{a_1 \tilde{x}_2 \tilde{x}_4+a_3}{a_2 \tilde{x}_2 \tilde{x}_4+a_1} \right) \;, \ \tilde{x}_3=\frac{1}{x_3}\left( \frac{a_1 \tilde{x}_2 \tilde{x}_4+a_3}{a_2 \tilde{x}_2 \tilde{x}_4+a_1} \right) \;. \end{aligned}$$

So, according to Theorem 1.3, the map \(\varphi :{\textbf{x}}\mapsto \tilde{{\textbf{x}}}\) preserves the two form

$$\begin{aligned} \omega =\frac{1}{x_1 x_2} \textrm{d}x_1 \wedge \textrm{d}x_2+\frac{1}{x_1 x_4} \textrm{d}x_1 \wedge \textrm{d}x_4-\frac{1}{x_2 x_3} \textrm{d}x_2 \wedge \textrm{d}x_3 +\frac{1}{x_3 x_4} \textrm{d}x_3 \wedge \textrm{d}x_4\;. \end{aligned}$$

In this case, the map \(\varphi \) corresponds to the (2, 2) periodic reduction in the lattice sine-Gordon equation (3.1) (see Fig. 6).

Fig. 6
figure 6

The (2, 2) staircase periodic reduction in the quadrilateral equation (3.1)

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The matrix B (and hence \({\omega }\)) is degenerate, of rank two. To obtain a symplectic map, we take a pair of monomials corresponding to an integer basis for

$$\begin{aligned} \textrm{im}\, B=<(1,0,1,0)^T,(0,1,0,1)^T>, \end{aligned}$$

namely

$$\begin{aligned} \pi : \qquad y_1=x_1x_3, \quad y_2=x_2x_4. \end{aligned}$$

Under the projection \(\pi \) defined above, \({\omega }\) is the pullback of the symplectic form

$$\begin{aligned} \hat{{\omega }}=\frac{1}{y_1y_2}\,\textrm{d}y_1\wedge \textrm{d}y_2, \end{aligned}$$

which is preserved by the induced map

$$\begin{aligned} \hat{\varphi }: \quad \left( \begin{array}{c} y_1 \\ y_2 \end{array}\right) \mapsto \left( \begin{array}{c}\tilde{y}_1 \\ \tilde{y}_2 \end{array} \right) , \quad \tilde{y}_2=\frac{1}{y_2}\left( \frac{a_1y_1+a_3}{a_2y_1+a_1}\right) ^2, \tilde{y}_1=\frac{1}{y_1}\left( \frac{a_1\tilde{y}_2+a_3}{a_2\tilde{y}_2+a_1}\right) ^2. \end{aligned}$$
(3.2)

The above map has the first integral

$$\begin{aligned} K=\frac{a_2^2y_1^2y_2^2+2a_1a_2(y_1^2y_2+y_1y_2^2)+a_1^2(y_1^2+y_2^2)+2a_1a_3(y_1+y_2)+a_3^2}{y_1y_2}, \end{aligned}$$

so it is Liouville integrable. In fact it is of QRT type: the level sets \(K=\,\)const are symmetric biquadratic curves, and \(\hat{\varphi }=\iota _h \circ \iota _v =(\iota \circ \iota _v)^2\) where the involutions \(\iota _h,\iota _v\) correspond to the horizontal and vertical switches on each level set, and \(\iota :\,y_1\leftrightarrow y_2\). For Laurentification of symmetric QRT maps, see [17].

In four dimensions, the other degrees of freedom in the original map \(\varphi \) have essentially trivial dynamics, since

$$\begin{aligned} \frac{\tilde{x}_1}{\tilde{x}_3}=\left( \frac{x_1}{x_3}\right) ^{-1}, \qquad \frac{\tilde{x}_2}{\tilde{x}_4}=\left( \frac{x_2}{x_4}\right) ^{-1}. \end{aligned}$$

3.2 \((3,-1)\) Periodic reduction in the lattice sine-Gordon equation

We consider the quiver with exchange matrix

$$\begin{aligned} B= \left( \begin{array}{rrrr} 0 &{} \quad 1 &{}\quad 0 &{}\quad 1 \\ -1 &{} \quad 0 &{}\quad 1 &{}\quad 0 \\ 0 &{} \quad -1 &{}\quad 0 &{} \quad 1 \\ -1 &{} \quad 0 &{}\quad -1 &{}\quad 0 \end{array} \right) . \end{aligned}$$

The matrix B is nondegenerate and satisfies \(\mu _1(B)=\rho (B)\) for the cyclic permutation \(\rho :(1,2,3,4)\mapsto (4,1,2,3)\), so it defines a cluster mutation-periodic quiver with period 1 [13]. Following the example in Sect. 3.1, we consider

$$\begin{aligned} g_1(x)=x\,\left( \frac{a_1 +a_3 x}{a_2 +a_1x}\right) \;. \end{aligned}$$

Here, \(M^+_1={x_2 x_4}\), \(M^-_1=1\) and

$$\begin{aligned} f_1(M^+_1,M^-_1)=M^+_1 g_1\left( \frac{M^-_1 }{M^+_1 }\right) = \frac{a_1 x_2 x_4+a_3}{a_2 x_2 x_4+a_1} . \end{aligned}$$

Hence, the appropriate analogue of Theorem 1.3 (see Remark 1.4) implies that the map \(\varphi =\rho ^{-1}\mu _1\) given by

$$\begin{aligned} \varphi :\, (x_1, x_2,x_3,x_4) \mapsto \left( x_2, x_3,x_{4},\frac{1}{x_1}\Big ( \frac{a_1 x_2 x_4+a_3}{a_2 x_2 x_4+a_1} \Big ) \right) \end{aligned}$$
(3.3)

preserves the symplectic form

$$\begin{aligned} \omega =\frac{1}{x_1 x_2} \textrm{d}x_1 \wedge \textrm{d}x_2+\frac{1}{x_1 x_4} \textrm{d}x_1 \wedge \textrm{d}x_4 +\frac{1}{x_2 x_3} \textrm{d}x_2 \wedge \textrm{d}x_3 +\frac{1}{x_3 x_4} \textrm{d}x_3 \wedge \textrm{d}x_4\;. \end{aligned}$$

The map (3.3) is associated with the \((3,-1)\) periodic reduction of the lattice sine-Gordon equation (3.1) and can be rewritten in recurrence form as

$$\begin{aligned} a_1(x_{n} x_{n+4}-x_{n+1}x_{n+3})+a_2x_{n}x_{n+1}x_{n+3}x_{n+4}=a_3\;. \end{aligned}$$

Closed-form expressions for integrals of periodic reductions in the sine-Gordon equation were presented in [34] and their involutivity was proved in [33].

4 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 in 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

$$\begin{aligned} a_k = \frac{y_k}{1+y_k}, \quad b_k=\frac{1}{1+y_k} \end{aligned}$$

in (2.1) leads to the expression for a mutation \(\mu _k\) in a cluster algebra with coefficients [11], which themselves mutate according to

$$\begin{aligned} y_j'={\left\{ \begin{array}{ll} y_k^{-1} &{} \text {if} \,\, j=k, \\ y_j \left( 1+y_k^{-\text {sgn}(b_{jk})} \right) ^{-b_{jk}} &{} \text {otherwise}. \end{array}\right. } \end{aligned}$$

The dynamics of the coefficients generates the associated Y-system [24]. In [20], 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

$$\begin{aligned} y_{n+7}y_n= \frac{ (1+y_{n+6})(1+y_{n+1})}{(1+y_{n+4}^{-1})(1+y_{n+3}^{-1})} \end{aligned}$$

corresponding to the Somos-7 recurrence (2.4), solved in terms of the q-Painlevé V equation

$$\begin{aligned} x_{n+2}x_n =x_{n+1}+{\alpha }_n\,{\mathfrak {q}}^n, \qquad {\alpha }_{n+6}={\alpha }_n, \end{aligned}$$
(4.1)

which is a non-autonomous version of the Lyness recurrence. The fact that the period of \({\alpha }_n\) is 6 is important, since if \({\mathfrak {q}}=1\) and \({\alpha }_n\) 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 \(g_1(x)=\frac{a x+b}{c x+d}\) and letting the coefficients a, b, c, d depend on the index n gives the sequence of symplectic maps

$$\begin{aligned} \varphi _n (x,y)=\left( y,\frac{a_n y+b_n}{x(c_n y+d_n)}\right) \end{aligned}$$

that corresponds to the non-autonomous nonlinear recurrence

$$\begin{aligned} x_{n+2}=\frac{a_n x_{n+1}+b_n}{x_n(c_n x_{n+1}+d_n)}\;. \end{aligned}$$

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 [30, 31].