On Torus Homeomorphisms Semiconjugate to irrational Rotations

TL;DR

This paper investigates torus homeomorphisms semiconjugate to irrational rotations, characterizing their structure via invariant foliations and providing criteria for the uniqueness of their rotation vectors, with implications for the Franks-Misiurewicz Conjecture.

Contribution

It characterizes these homeomorphisms through invariant foliations and extends Herman's results to establish criteria for rotation vector uniqueness.

Findings

Characterization of maps via invariant foliations with irrational combinatorics

Criteria for rotation vector uniqueness based on topological properties

Simplified proof of rotation vector uniqueness on certain annular continua

Abstract

In the context of the Franks-Misiurewicz Conjecture, we study homeomorphisms of the two-torus semiconjugate to an irrational rotation of the circle. As a special case, this conjecture asserts uniqueness of the rotation vector in this class of systems. We first characterise these maps by the existence of an invariant foliation by essential annular continua (essential subcontinua of the torus whose complement is an open annulus) which are permuted with irrational combinatorics. This result places the considered class close to skew products over irrational rotations. Generalising a well-known result of M. Herman on forced circle homeomorphisms, we provide a criterion, in terms of topological properties of the annular continua, for the uniqueness of the rotation vector. As a byproduct, we obtain a simple proof for the uniqueness of the rotation vector on decomposable invariant annular continua with empty interior. In addition, we collect a number of observations on the topology and rotation intervals of invariant annular continua with empty interior.