Periodic points and smooth rays

Abstract:

Let $P: \mathbb {C} \to \mathbb {C}$ be a polynomial map with disconnected filled Julia set $K_P$ and let $z_0$ be a repelling or parabolic periodic point of $P$. We show that if the connected component of $K_P$ containing $z_0$ is non-degenerate, then $z_0$ is the landing point of at least one smooth external ray. The statement is optimal in the sense that all but one cycle of rays landing at $z_0$ may be broken.