Hausdorff dimension of Julia sets of unicritical correspondences

TL;DR

This paper investigates the Hausdorff dimension of Julia sets for unicritical correspondences, establishing bounds via pressure functions and showing zero Lebesgue measure for certain parameter ranges.

Contribution

It provides a new upper bound for the Hausdorff dimension of Julia sets of unicritical correspondences using pressure functions and analyzes measure properties near zero parameters.

Findings

Hausdorff dimension bounded by zero of pressure function

Julia sets have zero Lebesgue measure near zero parameters

Results apply when $eta=p/q$ with $q^2<p$

Abstract

We show that if $\beta>1$ is a rational number and the Julia set $J$ of the holomorphic correspondence $z^{\beta}+c$ is a locally eventually onto hyperbolic repeller, then the Hausdorff dimension of $J$ is bounded from above by the zero of the associated pressure function. As a consequence, we conclude that the Julia set of the correspondence has zero Lebesgue measure for parameters close to zero, whenever $q^2<p$ and $\beta=p/q$ in lowest terms.