Discrete Dynamical Systems From Real Valued Mutation

TL;DR

This paper introduces a new family of discrete dynamical systems that generalize mutation dynamics in rank two cluster algebras, revealing properties like integrability, conserved quantities, and periodicity in special cases.

Contribution

It extends mutation dynamics to a broader class of systems, connecting tropical dynamics with matrix mutation and explaining asymptotic sign-coherence.

Findings

Systems exhibit integrability with preserved symplectic form

Existence of conserved quantities in tropical case

Certain cases show periodic tropical maps

Abstract

We introduce a family of discrete dynamical systems which includes, and generalizes, the mutation dynamics of rank two cluster algebras. These systems exhibit behavior associated with integrability, namely preservation of a symplectic form, and in the tropical case, the existence of a conserved quantity. We show in certain cases that the orbits are unbounded. The tropical dynamics are related to matrix mutation, from the theory of cluster algebras. We are able to show that in certain special cases, the tropical map is periodic. We also explain how our dynamics imply the asymptotic sign-coherence observed by Gekhtman and Nakanishi in the $2$-dimensional situation.