Continua as minimal sets of homeomorphisms of S^2

TL;DR

This paper studies minimal sets of orientation-preserving homeomorphisms on the 2-sphere, revealing invariant components with irrational rotation numbers and wandering others, advancing understanding of dynamical behavior on surfaces.

Contribution

It characterizes the structure of minimal sets on the sphere, showing exactly two invariant components and irrational rotation numbers, which is a novel insight in surface dynamics.

Findings

Exactly two invariant components of the complement of the minimal set

All other components are wandering under the homeomorphism

Invariant components have irrational Carathéodory rotation numbers

Abstract

Let $f$ be an orientation preserving homeomorphism of $S^2$ which has a (nontrivial) continuum $X$ as a minimal set. Then there are exactly two connected components of $S^2\setminus X$ which are left invariant by $f$ and all the others are wandering. The Carath\'eodory rotation number of an invariant component is irrational.