Symplectic Maps from Cluster Algebras

TL;DR

This paper explores the connection between cluster algebra mutations and symplectic maps, classifying integrable cases using algebraic entropy and illustrating examples of both integrable and non-integrable maps.

Contribution

It introduces a symplectic structure to cluster algebra-derived maps and classifies integrable cases through algebraic entropy analysis.

Findings

Symplectic structures are preserved under certain cluster mutation maps.

A classification of integrable versus non-integrable maps is achieved using algebraic entropy.

Examples demonstrate both integrable and non-integrable behaviors in these maps.

Abstract

We consider nonlinear recurrences generated from the iteration of maps that arise from cluster algebras. More precisely, starting from a skew-symmetric integer matrix, or its corresponding quiver, one can define a set of mutation operations, as well as a set of associated cluster mutations that are applied to a set of affine coordinates (the cluster variables). Fordy and Marsh recently provided a complete classification of all such quivers that have a certain periodicity property under sequences of mutations. This periodicity implies that a suitable sequence of cluster mutations is precisely equivalent to iteration of a nonlinear recurrence relation. Here we explain briefly how to introduce a symplectic structure in this setting, which is preserved by a corresponding birational map (possibly on a space of lower dimension). We give examples of both integrable and non-integrable maps that arise from this construction. We use algebraic entropy as an approach to classifying integrable cases. The degrees of the iterates satisfy a tropical version of the map.