Hyperbolic dimension and radial Julia sets of transcendental functions

TL;DR

This paper explores the relationship between the Hausdorff dimension of the reduced Julia set and the hyperbolic dimension for meromorphic functions, providing a simple proof of their equality.

Contribution

It offers a straightforward proof that the Hausdorff dimension of the reduced Julia set equals the hyperbolic dimension for Ahlfors islands maps.

Findings

Hausdorff dimension of the reduced Julia set equals hyperbolic dimension

Provides a simple proof of the dimension equality

Applicable to Ahlfors islands maps

Abstract

We survey the definition of the radial Julia set of a meromorphic function (in fact, more generally, any "Ahlfors islands map"), and give a simple proof that the Hausdorff dimension of the reduced Julia set always coincides with the hyperbolic dimension.