Dynamics and periodicity in a family of cluster maps

TL;DR

This paper analyzes the dynamics of a family of 4-dimensional cluster maps associated with mutation-periodic quivers, revealing three distinct behaviors based on a parameter, and provides a detailed description of their periodic points and orbit structures.

Contribution

It introduces a novel approach using presymplectic reduction to analyze the dynamics of cluster maps, classifying behaviors based on parameter values and describing their orbit structures.

Findings

Three types of dynamical behavior identified for different parameter values

Complete description of periodic points and orbit structures for each type

Use of first integrals to provide a detailed dynamical analysis

Abstract

The dynamics of a 1-parameter family of cluster maps $\varphi_r$ associated to mutation-periodic quivers in dimension 4, is studied in detail. The use of presymplectic reduction leads to a globally periodic symplectic map, and this enables us to reduce the problem to the study of maps belonging to a group of symplectic birational maps of the plane which is isomorphic to $SL(2,\mathbb{Z})\ltimes\mathbb{R}^2$. We conclude that there are three different types of dynamical behaviour for $\varphi_r$ characterized by the integer parameter values $r=1$, $r=2$ and $r>2$. For each type, the periodic points, the structure and the asymptotic behaviour of the orbits are completely described. A finer description of the dynamics is provided by using first integrals.