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

2020/11/09 by Park, Insung, Wu, Angela
#Dynamical Systems (math.DS) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2011.04150

Abstract

Suppose X is a compact connected metric space and f: X → 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 J_f is quasi-symmetric equivalent to a space having Hausdorff dimension 1 if and only if J_f is homeomorphic to a circle or a closed interval.

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