Hausdorff dimension of escaping sets of meromorphic functions

TL;DR

This paper characterizes the possible Hausdorff dimensions of escaping sets for meromorphic functions with finitely many singular values, demonstrating the full range from 0 to 2 and the diversity within quasiconformal classes.

Contribution

It constructs meromorphic functions with prescribed Hausdorff dimensions of escaping sets and shows uncountably many quasiconformally equivalent functions can have different dimensions.

Findings

For any d in [0,2], there exists a meromorphic function with escaping set dimension d.

The Hausdorff dimension of escaping sets can vary within quasiconformal classes.

The method involves gluing functions via quasiconformal mappings.

Abstract

We give a complete description of the possible Hausdorff dimensions of escaping sets for meromorphic functions with a finite number of singular values. More precisely, for any given $d\in [0,2]$ we show that there exists such a meromorphic function for which the Hausdorff dimension of the escaping set is equal to $d$. The main ingredient is to glue together suitable meromorphic functions by using quasiconformal mappings. Moreover, we show that there are uncountably many quasiconformally equivalent meromorphic functions for which the escaping sets have different Hausdorff dimensions.