Hausdorff dimension and biaccessibility for polynomial Julia sets

TL;DR

This paper studies the set of biaccessible points in polynomial Julia sets, proving that their external angles' Hausdorff dimension is less than 1 unless the set is an interval, extending previous measure-based results.

Contribution

It establishes a Hausdorff dimension bound for biaccessible points in polynomial Julia sets, generalizing earlier measure-based theorems to a dimension context.

Findings

Hausdorff dimension of biaccessible external angles is less than 1 for non-interval Julia sets

The result extends previous measure-zero findings to Hausdorff dimension

Theorem applies to polynomial Julia sets of any degree d ≥ 2

Abstract

We investigate the set of biaccessible points for connected polynomial Julia sets of arbitrary degrees $d\geq 2$. We prove that the Hausdorff dimension of the set of external angles corresponding to biaccessible points is less than 1, unless the Julia set is an interval. This strengthens theorems of Stanislav Smirnov and Anna Zdunik: they proved that the same set of external angles has zero 1-dimensional measure.