Hausdorff limits of external rays: the topological picture

TL;DR

This paper investigates how external rays of a fixed periodic angle behave in the limit as polynomials of degree at least two with connected Julia sets converge, revealing the topological structure of their Hausdorff limits.

Contribution

It provides a detailed topological description of the Hausdorff limits of external rays for converging polynomial sequences with connected Julia sets.

Findings

Hausdorff limits of external rays are characterized topologically.

The limits depend on the convergence of the polynomial sequence.

The results apply to polynomials of degree at least two with connected Julia sets.

Abstract

We study Hausdorff limits of the external rays of a given periodic angle along a convergent sequence of polynomials of degree $d \geq 2$ with connected Julia sets.