The Franks-Misiurewicz conjecture for extensions of irrational rotations

TL;DR

This paper proves that certain toral homeomorphisms semiconjugate to irrational rotations are pseudo-rotations, advancing the understanding of the Franks-Misiurewicz conjecture in minimal dynamical systems.

Contribution

It establishes that such homeomorphisms have a singleton rotation set, completing the minimal case analysis of the conjecture.

Findings

Homeomorphisms homotopic to identity and semiconjugate to irrational rotations are pseudo-rotations.

Complete the minimal case of the Franks-Misiurewicz conjecture.

Supports the conjecture that certain toral dynamics have singleton rotation sets.

Abstract

We show that a toral homeomorphism which is homotopic to the identity and topologically semiconjugate to an irrational rotation of the circle is always a pseudo-rotation (i.e. its rotation set is a single point). In combination with recent results, this allows us to complete the study of the Franks-Misiurewicz conjecture in the minimal case.