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

Angela Wu; Insung Park

doi:10.3934/dcds.2023138

Article Contents

2024, Volume 44, Issue 4: 1073-1086. Doi: 10.3934/dcds.2023138

This issue Previous Article Kinetic Schauder estimates with time-irregular coefficients and uniqueness for the Landau equation Next Article A class of critical $ p $-Kirchhoff type equations on closed manifolds

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

Angela Wu^1, , and
Insung Park^2,

1.
Department of Mathematics, Indiana University, Bloomington, IN 47405 USA
2.
The Institute for Computational and Experimental Research in Mathematics, Providence, RI, 02903 USA

^*Corresponding author: Angela Wu
^*Corresponding author: Angela Wu

Received: April 2023

Revised: September 2023

Early access: November 20, 2023

Published online: November 20, 2023

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Abstract

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.

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Mathematics Subject Classification: 37F35, 37F10.

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References

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