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

Suppose $ X $ is a compact connected metric space and $ f\colon 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. [ABSTRACT FROM AUTHOR]