Hausdorff dimension of escaping sets of meromorphic functions

Magnus Aspenberg, Weiwei Cui

Published 2020-04-23Version 1

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 in the sense of Eremenko and Lyubich for which the escaping sets have different Hausdorff dimensions.