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 $X$.