Accessible points rotate as prime ends in backward or forward time

TL;DR

This paper investigates the rotation behavior of accessible points in a specific annular invariant set, establishing conditions under which their rotation numbers align with the prime end rotation number in either forward or backward time.

Contribution

It proves that accessible points' rotation numbers in a certain annular invariant set are well-defined and match the prime end rotation number in either forward or backward time, depending on the case.

Findings

Accessible points' rotation numbers are well-defined.

Rotation numbers match the prime end rotation number.

Either forward or backward semi-orbits share this property.

Abstract

Let $X$ be a closed invariant subset of the half--open annulus $\mathbb{A}$ such that $\mathbb{A} \setminus X$ is homeomorphic to $\mathbb{A}$. We prove that either the rotation number of all forward semi--orbits of accessible points of $X$ are well--defined and equal to the prime end rotation number or the same is true for all backward semi--orbits of accessible points of $X$.