The Hausdorff dimension of the boundary of the immediate basin of infinity of McMullen maps

TL;DR

This paper derives a formula for the Hausdorff dimension of the boundary of the immediate basin of infinity in McMullen maps, providing insights into the fractal geometry of their Julia sets.

Contribution

It introduces a new formula for the Hausdorff dimension of the boundary of the immediate basin of infinity in McMullen maps with small parameters.

Findings

Provides a lower bound for the Hausdorff dimension of Julia sets

Derives an explicit formula for the boundary's Hausdorff dimension

Enhances understanding of fractal structures in complex dynamics

Abstract

In this paper, we give a formula of the Hausdorff dimension of the boundary of the immediate basin of infinity of McMullen maps $f_p(z)=z^Q+p/z^Q$, where $Q\geq 3$ and $p$ is small. This gives a lower bound of the Hausdorff dimension of the Julia sets of McMullen maps in the special cases.