Periodic Points and Smooth Rays

TL;DR

This paper proves that for certain polynomial maps with disconnected Julia sets, each non-degenerate component containing a repelling or parabolic periodic point is the landing point of at least one smooth external ray, with some rays possibly broken.

Contribution

It establishes the existence of smooth external rays landing at specific periodic points in disconnected Julia sets, extending understanding of the geometric structure of such sets.

Findings

At least one smooth external ray lands at each relevant periodic point.

All but possibly one ray landing at a point may be broken.

The result is optimal, with examples showing the possibility of broken 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 {\it smooth} external ray. The statement is optimal in the sense that all but one ray landing at $z_0$ may be broken.