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arXiv · 1101.4209

Brushing the hairs of transcendental entire functions

Krzysztof Barański·Xavier Jarque·Lasse Rempe
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Abstract

Let f be a hyperbolic transcendental entire function of finite order in the Eremenko-Lyubich class (or a finite composition of such maps), and suppose that f has a unique Fatou component. We show that the Julia set of $f$ is a Cantor bouquet; i.e. is ambiently homeomorphic to a straight brush in the sense of Aarts and Oversteegen. In particular, we show that any two such Julia sets are ambiently homeomorphic. We also show that if $f\in\B$ has finite order (or is a finite composition of such maps), but is not necessarily hyperbolic, then the Julia set of f contains a Cantor bouquet. As part of our proof, we describe, for an arbitrary function $f\in\B$, a natural compactification of the dynamical plane by adding a "circle of addresses" at infinity.

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BibTeXRIS

Krzysztof Barański, Xavier Jarque, Lasse Rempe. 2012-01-26. Brushing the hairs of transcendental entire functions. https://doi.org/10.1016/j.topol.2012.02.004

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