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

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 Haïssinsky-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.