Recurrence of non-resonant homeomorphisms on the torus

TL;DR

This paper proves that certain torus homeomorphisms with a single irrational rotation vector are chain-recurrent and have weakly transitive nonwandering sets, with an example showing non-transitivity can occur.

Contribution

It establishes chain-recurrence and weak transitivity for non-resonant torus homeomorphisms, and provides an example of non-transitive nonwandering sets.

Findings

Homeomorphisms with a single irrational rotation vector are chain-recurrent.

Pseudo-orbits can be constructed with few jumps.

Nonwandering set may not be transitive.

Abstract

We prove that a homeomorphism of the torus homotopic to the identity whose rotation set is reduced to a single totally irrational vector is chain-recurrent. In fact, we show that pseudo-orbits can be chosen with a small number of jumps, in particular, that the nonwandering set is weakly transitive. We give an example showing that the nonwandering set of such a homeomorphism may not be transitive.