On the Hausdorff dimension of the Sierpiński Julia sets

Krzysztof Barański; Michał Wardal

doi:10.3934/dcds.2015.35.3293

Article Contents

2015, Volume 35, Issue 8: 3293-3313. Doi: 10.3934/dcds.2015.35.3293

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On the Hausdorff dimension of the Sierpiński Julia sets

Krzysztof Barański^1, and
Michał Wardal^1,

1.
Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa

Received: February 28, 2014

Revised: November 30, 2014

Published: May 2017

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Abstract

We estimate the Hausdorff dimension of hyperbolic Julia sets of maps from the well-known family $F_{\lambda,n}(z) = z^n + \lambda/z^n$, $n \ge 2$, $\lambda \in \mathbb{C} \setminus \{0\}$. In particular, we show that $\dim_H J(F_{\lambda,n}) = \mathcal O (1/\ln |\lambda|)$ for large $|\lambda|$, and $\dim_H J(F_{\lambda,n}) = 1 + \mathcal O (1/\ln n)$ for large $n$ in the three cases: when $J(F_{\lambda,n})$ is a Cantor set, a Cantor set of quasicircles and a Sierpiński curve.

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Mathematics Subject Classification: Primary: 37F10, 37F35; Secondary: 28A78, 28A80.

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