Rotation sets of invariant separating continua of annular homeomorphisms

TL;DR

This paper investigates the rotation sets of invariant continua in annular homeomorphisms, establishing the existence of periodic points for rational rotation numbers and relating Carathéodory rotation numbers to the rotation set.

Contribution

It provides new conditions under which rational rotation numbers guarantee periodic points and links Carathéodory rotation numbers to the rotation set for invariant continua.

Findings

Existence of periodic points for rational rotation numbers under specified conditions.

Carathéodory rotation numbers of boundary domains are contained in the rotation set.

Conditions for invariant continua to have rotation sets with specific properties.

Abstract

Let $f$ be a homeomorphism of the closed annulus $A$ isotopic to the identity, and let $X\subset {\rm Int}A$ be an $f$-invariant continuum which separates $A$ into two domains, the upper domain $U_+$ and the lower domain $U_-$. Fixing a lift of $f$ to the universal cover of $A$, one defines the rotation set $\tilde \rho(X)$ of $X$ by means of the invariant probabilities on $X$. For any rational number $p/q\in \tilde\rho(X)$, $f$ is shown to admit a $p/q$ periodic point in $X$, provided that (1) $X$ consists of nonwandering points or (2) $X$ is an attractor and the frontiers of $U_-$ and $U_+$ coincides with $X$. Also the Carath\'eodory rotation numbers of $U_\pm$ are shown to be in $\tilde\rho(X)$ for any separating invariant continuum $X$.