A triple boundary lemma for surface homeomorphisms

TL;DR

This paper proves a fixed point property for certain boundary points under area-preserving surface homeomorphisms and applies it to invariant continua and disks, providing new insights into surface dynamics.

Contribution

It introduces a triple boundary lemma for surface homeomorphisms and applies it to invariant sets, offering new elementary proofs and fixed point results.

Findings

Points on the boundary of three disjoint invariant disks are fixed points.

Invariant Wada continua are periodic under the homeomorphism.

Invariant disks are homotopically bounded if fixed points are inessential.

Abstract

Given an orientation-preserving and area-preserving homeomorphism $f$ of the sphere, we prove that every point which is in the common boundary of three pairwise disjoint invariant open topological disks must be a fixed point. As an application, if $K$ is an invariant Wada type continuum, then $f^n|_K$ is the identity for some $n>0$. Another application is an elementary proof of the fact that invariant disks for a nonwandering homeomorphisms homotopic to the identity in an arbitrary surface are homotopically bounded if the fixed point set is inessential. The main results in this article are self-contained.