Aperiodic invariant continua for surface homeomorphisms

TL;DR

This paper characterizes invariant continua for surface homeomorphisms without wandering points, showing they are either the entire surface (on tori) or intersections of annuli, extending results to non-orientable surfaces.

Contribution

It provides a classification of invariant continua for surface homeomorphisms with no wandering points, including non-orientable cases, and describes their structure.

Findings

If K is invariant and contains no periodic points, then K=S on a torus.

K is the intersection of a decreasing sequence of annuli.

The results extend to non-orientable surfaces.

Abstract

We prove that if a homeomorphism of a closed orientable surface S has no wandering points and leaves invariant a compact, connected set K which contains no periodic points, then either K=S and S is a torus, or $K$ is the intersection of a decreasing sequence of annuli. A version for non-orientable surfaces is given.