Hausdorff dimensions of escaping sets of transcendental entire functions

Rempe, Lasse; Stallard, Gwyneth

doi:10.1090/S0002-9939-09-10104-1

Hausdorff dimensions of escaping sets of transcendental entire functions
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by Lasse Rempe and Gwyneth M. Stallard PDF

Proc. Amer. Math. Soc. 138 (2010), 1657-1665 Request permission

Abstract:

Suppose that $f$ and $g$ are transcendental entire functions, each with a bounded set of singular values, and that $g\circ \phi = \psi \circ f$, where $\phi ,\psi :\mathbb {C}\to \mathbb {C}$ 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.

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