On the Hausdorff dimension of the Julia set of a regularly growing entire function

We show that if the growth of a transcendental entire function f is sufficiently regular, then the Julia set and the escaping set of f have Hausdorff dimension 2.Comment: 21 page

On the dimension of points which escape to infinity at given rate under exponential iteration

We prove a number of results concerning the Hausdorff and packing dimension of sets of points which escape (at least in average) to infinity at a given rate under non-autonomous iteration of exponential maps. In particular, we generalize the results proved by Sixsmith in 2016 and answer his question on annular itineraries for exponential maps

Absorbing sets and Baker domains for holomorphic maps

We consider holomorphic maps $f: U \to U$ for a hyperbolic domain $U$ in the complex plane, such that the iterates of $f$ converge to a boundary point $\zeta$ of $U$. By a previous result of the authors, for such maps there exist nice absorbing domains $W \subset U$. In this paper we show that $W$ can be chosen to be simply connected, if $f$ has parabolic I type in the sense of the Baker--Pommerenke--Cowen classification of its lift by a universal covering (and $\zeta$ is not an isolated boundary point of $U$). Moreover, we provide counterexamples for other types of the map $f$ and give an exact characterization of parabolic I type in terms of the dynamical behaviour of $f$

Dimension properties of the boundaries of exponential basins

We prove that the boundary of a component $U$ of the basin of an attracting periodic cycle (of period greater than 1) for an exponential map on the complex plane has Hausdorff dimension greater than 1 and less than 2. Moreover, the set of points in the boundary of $U$ which do not escape to infinity has Hausdorff dimension (in fact: hyperbolic dimension) greater than 1, while the set of points in the boundary of $U$ which escape to infinity has Hausdorff dimension 1.Comment: 13 pages, 1 figur

The growth rate of an entire function and the Hausdorff dimension of its Julia set

Let f be a transcendental entire function in the Eremenko-Lyubich class B. We give a lower bound for the Hausdorff dimension of the Julia set of f that depends on the growth of f. This estimate is best possible and is obtained by proving a more general result concerning the size of the escaping set of a function with a logarithmic tract.Comment: 19 page