A characterization of annularity for area-preserving toral homeomorphisms

TL;DR

This paper establishes conditions under which area-preserving toral homeomorphisms with specific rotation sets possess essential invariant annuli, linking rotation set geometry to topological dynamics.

Contribution

It proves that a nondegenerate vertical rotation set containing zero implies the existence of an essential invariant annulus for such homeomorphisms.

Findings

Existence of an essential invariant annulus under given rotation set conditions

Lift to universal cover has bounded horizontal displacement

Rotation set geometry influences topological structure

Abstract

We prove that if an area-preserving homeomorphism of the torus in the homotopy class of the identity has a rotation set which is a nondegenerate vertical segment containing the origin, then there exists an essential invariant annulus. In particular, some lift to the universal covering has uniformly bounded displacement in the horizontal direction.