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

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

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

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

Lebesgue Measure of Escaping Sets of Entire Functions in the Eremenko-Lyubich Class

The main aim of this thesis is to try to understand dynamical behaviours of transcendental entire functions from a measure-theoretic point of view. More precisely, we concentrate on the Eremenko-Lyubich class \b, and study the measure properties of the Julia and escaping sets. For a transcendental entire function, the complex plane is partitioned into the Fatou and Julia sets based on the theory of normal families. Various properties of the Fatou and Julia sets have been studied for a long time. Another interesting set which is of equal importance is the escaping set, consisting of points in \c tending to infinity under iterates. For entire functions in class \b, the escaping set turns out to be a subset of the Julia set. Thus this gives us a way to estimate the size of Julia sets from below. In 1987, McMullen showed that the Julia set of $\lambda e^z$ has Hausdorff dimension $2$ and the area of Julia set of $\sin(\alpha z+\beta)$ is positive, where $\lambda\in\c\setminus\{0\}$, \alpha,\beta\in\c and $\alpha\neq 0$. The result on the Hausdorff dimension has been extended to more general transcendental entire functions by various authors, while the generalization of the result on the area of Julia set to larger class of functions was not undertook until recent work of Aspenberg and Bergweiler. They gave a condition that is satisfied by many functions, in particular the ones considered by McMullen. We continue their work on this respect and show that their condition is essentially sharp by constructing an entire function for which the escaping set has zero area. In 1992, Eremenko and Lyubich gave a condition under which the area of escaping set of entire functions in the class \b is zero. The condition is quite general and in particular applies to finite order entire functions in class \b whose inverse has a finite logarithmic singularity. We shall generalize this result to certain functions of infinite order, by adapting the method we use above