Hausdorff dimensions of escaping sets of transcendental entire functions

TL;DR

This paper proves that affinely equivalent transcendental entire functions have escaping sets with identical Hausdorff dimensions and constructs a family of such functions with escaping sets of Hausdorff dimension one.

Contribution

It establishes the invariance of Hausdorff dimension of escaping sets under affine equivalence and provides examples with dimension one.

Findings

Escaping sets of affinely equivalent functions share the same Hausdorff dimension.

Existence of transcendental entire functions with escaping set Hausdorff dimension equal to one.

Hausdorff dimension invariance under affine conjugation for transcendental entire functions.

Abstract

Let $f$ and $g$ be transcendental entire functions, each with a bounded set of singular values, and suppose that $f$ and $g$ are affinely equivalent (that is, $g \circ \phi= \psi\circ f$, where $\phi,\psi:\C\to\C$ are affine). We show that the escaping sets of $f$ and $g$ have the same Hausdorff dimension. Using a result of the second author, we deduce that there exists a family of transcendental entire functions for which the escaping set has Hausdorff dimension equal to one.