Transitivity and the existence of horseshoes on the 2-torus

TL;DR

This paper explores how certain conditions like transitivity and homotopy type on the 2-torus imply the existence of chaotic dynamics such as horseshoes, linking topological properties to chaos.

Contribution

It establishes new conditions under which transitive homeomorphisms on the 2-torus guarantee topological horseshoes, connecting homotopy classes to chaos.

Findings

Transitive homeomorphisms homotopic to identity with fixed and non-fixed periodic points have horseshoes.

Transitive homeomorphisms homotopic to a Dehn twist are either aperiodic or have horseshoes.

Conditions linking topological properties to chaotic dynamics on the 2-torus.

Abstract

We study the relationship between transitivity and topological chaos for homeomorphisms of the two torus. We show that if a transitive homeomorphism of $\mathbb{T}^2$ is homotopic to the identity and has both a fixed point and a periodic point which is not fixed, then it has a topological horseshoe. We also show that if a transitive homeomorphims of $\mathbb{T}^2$ is homotopic to a Dehn twist, then either it is aperiodic or it has a topological horseshoe.