Discrete integrable systems and Poisson algebras from cluster maps

TL;DR

This paper studies cluster mutation-periodic quivers, their associated nonlinear recurrences, and demonstrates how these maps can possess invariant Poisson structures, leading to integrable discrete dynamical systems with low entropy.

Contribution

It classifies cluster maps with period 1, constructs invariant Poisson structures, and links their entropy to tropical analogues, identifying four families with zero entropy that are integrable.

Findings

Four families of cluster maps have zero entropy.

Explicit examples of integrable discrete systems are provided.

Entropy can be predicted from tropical analogues.

Abstract

We consider nonlinear recurrences generated from cluster mutations applied to quivers that have the property of being cluster mutation-periodic with period 1. Such quivers were completely classified by Fordy and Marsh, who characterised them in terms of the skew-symmetric matrix that defines the quiver. The associated nonlinear recurrences are equivalent to birational maps, and we explain how these maps can be endowed with an invariant Poisson bracket and/or presymplectic structure. Upon applying the algebraic entropy test, we are led to a series of conjectures which imply that the entropy of the cluster maps can be determined from their tropical analogues, which leads to a sharp classification result. Only four special families of these maps should have zero entropy. These families are examined in detail, with many explicit examples given, and we show how they lead to discrete dynamics that is integrable in the Liouville-Arnold sense.