Quasi-self-similar fractals containing "Y" have dimension larger than one

TL;DR

This paper proves that certain fractal spaces containing a 'Y' shape have a Hausdorff dimension greater than one, with implications for the structure of Julia sets of semi-hyperbolic rational maps.

Contribution

It establishes a link between the topological presence of a 'Y' shape in fractals and their Hausdorff dimension, extending understanding of fractal geometry and complex dynamics.

Findings

Spaces with a 'Y' shape have Hausdorff dimension > 1.

Julia sets homeomorphic to a circle or interval are quasi-symmetrically equivalent to dimension 1 spaces.

Presence of 'Y' shapes influences the fractal's dimensional properties.

Abstract

Suppose $X$ is a compact connected metric space and $f: X \to X$ is a metric coarse expanding conformal map in the sense of Ha\"issinsky-Pilgrim. We show that if $X$ contains a homeomorphic copy of the letter "Y", then the Hausdorff dimension of $X$ is greater than one. As an application, we show that for a semi-hyperbolic rational map $f$ its Julia set $\mathcal{J}_f$ is quasi-symmetric equivalent to a space having Hausdorff dimension 1 if and only if $\mathcal{J}_f$ is homeomorphic to a circle or a closed interval.