Hausdorff dimension of escaping sets of meromorphic functions

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 ∈ [ 0 , 2 ] 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 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.

The Hausdorff dimension of Julia sets of meromorphic functions in the Speiser class

“…This is due to Eremenko and Lyubich [24, Theorem 1] for entire f and to Rippon and Stallard [40, Theorem A] for meromorphic f . Theorem 1.1 complements the result of [2] where it was shown that for all d ∈ [0, 2] there exists a meromorphic function f ∈ S such that dim I(f ) = d.…”

The Hausdorff dimension of Julia sets of meromorphic functions in the Speiser class

“…This is due to Eremenko and Lyubich [24, Theorem 1] for entire f and to Rippon and Stallard [40, Theorem A] for meromorphic f . Theorem 1.1 complements the result of [2] where it was shown that for all d ∈ [0, 2] there exists a meromorphic function f ∈ S such that dim…”

“…As a consequence of our main result, we also have the following theorem which completes the study begun in [2] and continued in [3].…”

“…Our results can be seen as a contribution to recent efforts to understand the differences and similarities between the classes B and S. We refer to [2,3,10,15,16,17,21] and the survey [37] for results in this direction. As an example we mention the recent result of Albrecht and Bishop [1] who showed that given δ > 0 there exists an entire function f ∈ S such that 1 < dim J(f ) < 1 + δ.…”

“…Both the Speiser and the Eremenko-Lyubich class play an important role in transcendental dynamics. The similarities and differences between these classes are addressed in a number of recent papers [2,[12][13][14]21]. A survey of some of these as well as many other results concerning the dynamics of functions in S and B is given in [41].…”