Prime end rotation numbers of invariant separating contunua of annular homeomorphisms

TL;DR

This paper proves that the prime end rotation numbers of invariant separating continua in annular homeomorphisms are always contained within the continuum's rotation set, linking boundary dynamics to interior invariant sets.

Contribution

It establishes the inclusion of prime end rotation numbers in the rotation set for invariant separating continua in annular homeomorphisms, a new connection in dynamical systems.

Findings

Prime end rotation numbers belong to the rotation set of the continuum.

The result applies to any invariant separating continuum in the annulus.

Connects boundary rotation dynamics with interior invariant sets.

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 $X$, as well as the prime end rotation number $\check\rho_\pm$ of $U_\pm$. The purpose of this paper is to show that $\check\rho_\pm$ belongs to $\tilde\rho(X)$ for any separating invariant continuum $X$.