McMullen's and geometric pressures and approximating Hausdorff dimension of Julia sets from below

TL;DR

This paper introduces new variants of geometric pressure for rational functions, aiming to efficiently approximate the hyperbolic Hausdorff dimension of Julia sets from below, including for non-hyperbolic cases.

Contribution

It proposes novel geometric pressure variants that facilitate the approximation of Julia set dimensions, extending methods to non-hyperbolic rational functions.

Findings

New pressure variants for rational functions

Effective lower approximation of Julia set dimensions

Extension to non-hyperbolic functions

Abstract

We introduce new variants of the notion of geometric pressure for rational functions on the Riemann sphere, including non-hyperbolic functions, in the hope some of them occur useful to achieve a fast approximation from below of the hyperbolic Hausdorff dimension of Julia set.