Realizing rotation numbers on annular continua

TL;DR

This paper extends the understanding of rotation sets in invariant annular continua, showing they are closed intervals with all intermediate rational rotation numbers realized by periodic orbits, generalizing previous results.

Contribution

It adapts Handel's result to invariant annular continua, proving the rotation set is closed and all elements are realized, and shows minimal continua (circloids) have rotation sets as closed intervals with all points realized.

Findings

Rotation set in invariant annular continua is closed.

Every element of the rotation set is realized by an ergodic measure or periodic orbit.

In minimal annular continua, the rotation set is a closed interval with all points realized.

Abstract

An annular continuum is a compact connected set $K$ which separates a closed annulus $A$ into exactly two connected components, one containing each boundary component. The topology of such continua can be very intricate (for instance, non-locally connected). We adapt a result proved by Handel in the case where $K=A$, showing that if $K$ is an invariant annular continuum of a homeomorphism of $A$ isotopic to the identity, then the rotation set in $K$ is closed. Moreover, every element of the rotation set is realized by an ergodic measure supported in $K$ (and by a periodic orbit if the rotation number is rational) and most elements are realized by a compact invariant set. Our second result shows that if the continuum $K$ is minimal with the property of being annular (what we call a circloid), then every rational number between the extrema of the rotation set in $K$ is realized by a periodic orbit in $K$. As a consequence, the rotation set is a closed interval, and every number in this interval (rational or not) is realized by an orbit (moreover, by an ergodic measure) in $K$. This improves a previous result of Barge and Gillette.