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

TL;DR

This paper establishes a lower bound on the Hausdorff dimension of Julia sets for transcendental entire functions in class B, linking it to the functions' growth rate, and proves the bound's optimality.

Contribution

It introduces a new lower bound for the Hausdorff dimension of Julia sets based on the growth of functions in class B, extending understanding of their fractal geometry.

Findings

Lower bound depends on the growth of the entire function

Bound is proven to be optimal

Results apply to functions with a logarithmic tract

Abstract

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.