Dimensions of Julia sets of hyperbolic entire functions

Stallard, Gwyneth M. (2004). Dimensions of Julia sets of hyperbolic entire functions. Bulletin of the London Mathematical Society, 36(2) pp. 263–270.

DOI: https://doi.org/10.1112/S0024609303002698

URL: http://journals.cambridge.org/action/displayAbstra...

Abstract

It is known that, if $f$ is a hyperbolic rational function, then the Hausdorff, packing and box dimensions of the Julia set $J(f)$ are equal. It is also known that there is a family of hyperbolic transcendental meromorphic functions with infinitely many poles for which this result fails to be true. In this paper, new methods are used to show that there is a family of hyperbolic transcendental entire functions $f_K$ , $K \in \N$ , such that the box and packing dimensions of $J(f_K)$ are equal to two, even though as $K \to \infty$ the Hausdorff dimension of $J(f_K)$ tends to one, the lowest possible value for the Hausdorff dimension of the Julia set of a transcendental entire function.