Hausdorff dimension of escaping sets of meromorphic functions

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.Comment: 37 pages, 8 figures. Some overall revision in the introduction. More details added in Section

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

Bergweiler and Kotus gave sharp upper bounds for the Hausdorff dimension of the escaping set of a meromorphic function in the Eremenko-Lyubich class, in terms of the order of the function and the maximal multiplicity of the poles. We show that these bounds are also sharp in the Speiser class. We apply this method also to construct meromorphic functions in the Speiser class with preassigned dimensions of the Julia set and the escaping set.Comment: 26 pages; minor revision of v

Hausdorff dimension of escaping sets of meromorphic functions II

A function which is transcendental and meromorphic in the plane has at least two singular values. On one hand, if a meromorphic function has exactly two singular values, it is known that the Hausdorff dimension of the escaping set can only be either $2$ or $1/2$. On the other hand, the Hausdorff dimension of escaping sets of Speiser functions can attain every number in $[0,2]$ (cf. \cite{ac1}). In this paper, we show that number of singular values which is needed to attain every Hausdorff dimension of escaping sets is not more than $4$.Comment: 22 pages, 5 figure

Speiser meets Misiurewicz

We propose a notion of Misiurewicz condition for transcendental entire functions and study perturbations of Speiser functions satisfying this condition in their parameter spaces (in the sense of Eremenko and Lyubich). We show that every Misiurewicz entire function can be approximated by hyperbolic maps in the same parameter space. Moreover, Misiurewicz functions are Lebesgue density points of hyperbolic maps if their Julia sets have zero Lebesgue measure. We also prove that the set of Misiurewicz Speiser functions has Lebesgue measure zero in the parameter space.Comment: 37 pages, 1 figure. Author accepted version. Overall revision on Section 2 and Section 4. To appear in Communications in Mathematical Physic

Perturbations of exponential maps: Non-recurrent dynamics

We study perturbations of non-recurrent parameters in the exponential family. It is shown that the set of such parameters has Lebesgue measure zero. This particularly implies that the set of escaping parameters has Lebesgue measure zero, which complements a result of Qiu from 1994. Moreover, we show that non-recurrent parameters can be approximated by hyperbolic ones.Comment: Author accepted version. Overall revisions on Section 2 and simplified proofs for Lemmas 3.2 and 3.3; several references added. To appear in Journal d'Analyse Math\'ematiqu

Ergodic exponential maps with escaping singular behaviours

We construct exponential maps for which the singular value tends to infinity under iterates while the maps are ergodic. This is in contrast with a result of Lyubich from 1987 which tells that $e^z$ is not ergodic.Comment: 11 pages, 1 figure; comments are welcom

Slowly recurrent Collet-Eckmann maps with non-empty Fatou set

In this paper we study rational Collet-Eckmann maps for which the Julia set is not the whole sphere and for which the critical points are recurrent at a slow rate. In families where the orders of the critical points are fixed, we prove that such maps are Lebesgue density points of hyperbolic maps. In particular, if all critical points are simple, they are Lebesgue density points of hyperbolic maps in the full space of rational maps of any degree $d \geq 2$