The Hausdorff dimension of quasi-circles: a result of Ruelle and Bowen

TL;DR

This paper offers a clearer, more accessible proof of Bowen and Ruelle's famous formula for the Hausdorff dimension of Julia sets of quadratic polynomials with small parameters, aimed at graduate students.

Contribution

The paper provides an expanded, clarified proof of a key result relating to the Hausdorff dimension of quasi-circles, making it more accessible for students.

Findings

Provides an accessible proof of the Hausdorff dimension formula

Clarifies the relationship between Julia sets and quasi-circles

Reinforces the asymptotic behavior for small parameters

Abstract

We provide an expanded and clarified proof of the famous result of Bowen and Ruelle giving an asymptotic formula for the Hausdorff dimension of quasi-circles corresponding to the Julia sets of $f(z)=z^2+c$ for small $c$. The proof does not contain new material but has been rewritten to make it more accessible to MSc or PhD students with an interest in dimension theory.