Dynamics of shear homeomorphisms of tori and the Bestvina-Handel algorithm

TL;DR

This paper extends Sharkovskii's theorem to shear homeomorphisms of tori, establishing a dynamical order relation for periodic orbits and applying it to analyze a quantum chaotic system.

Contribution

It introduces a new order relation for periodic orbits of shear homeomorphisms of tori and applies it to a quantum chaotic physical system.

Findings

Established a Sharkovskii-type theorem for torus shear homeomorphisms

Developed a dynamical order relation for simple orbits

Applied the theory to analyze a kicked accelerated particle system

Abstract

Sharkovskii proved that the existence of a periodic orbit in a one-dimensional dynamical system implies existence of infinitely many periodic orbits. We obtain an analog of Sharkovskii's theorem for periodic orbits of shear homeomorphisms of the torus. This is done by obtaining a dynamical order relation on the set of simple orbits and simple pairs. We then use this order relation for a global analysis for a quantum chaotic physical system called the kicked accelerated particle.