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arXiv · 2011.04150

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

Insung Park·Angela Wu
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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.

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Insung Park, Angela Wu. 2020-11-09. Quasi-self-similar fractals containing "Y" have dimension larger than one. https://arxiv.org/abs/2011.04150

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