Periodic Points and Smooth Rays

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 {\it smooth} external ray. The statement is optimal in the sense that all but one ray landing at $z_0$ may be broken.Comment: 10 pages, 3 figure