Accessible points rotate as prime ends in backward or forward time

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 $X$.