Assouad type dimensions of parabolic Julia sets

TL;DR

This paper determines the Assouad dimension of parabolic Julia sets as the maximum of 1 and the Hausdorff dimension, revealing cases where these dimensions differ, and explores their spectra and measures.

Contribution

It computes all Assouad type dimensions for parabolic Julia sets and the associated h-conformal measure, and characterizes the Assouad dimension in presence of Cremer points.

Findings

Assouad dimension equals max{1, Hausdorff dimension} for parabolic Julia sets.

Assouad dimension can be 2 if the Julia set has a Cremer point.

Assouad and Hausdorff dimensions can differ significantly.

Abstract

We prove that the Assouad dimension of a parabolic Julia set is $\max\{1,h\}$ where $h$ is the Hausdorff dimension of the Julia set. Since $h$ may be strictly less than 1, this provides examples where the Assouad and Hausdorff dimension are distinct. The box and packing dimensions of the Julia set are known to coincide with $h$ and, moreover, $h$ can be characterised by a topological pressure function. This distinctive behaviour of the Assouad dimension invites further analysis of the Assouad type dimensions, including the Assouad and lower spectra. We compute all of the Assouad type dimensions for parabolic Julia sets and the associated $h$-conformal measure. Further, we show that if a Julia set has a Cremer point, then the Assouad dimension is 2.