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

Cui, Weiwei

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.

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Cui, Weiwei: Lebesgue Measure of Escaping Sets of Entire Functions in the Eremenko-Lyubich Class. 2017.

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