Boundaries of univalent Baker domains

Rippon, P. J. and Stallard, G. M. (2018). Boundaries of univalent Baker domains. Journal d'Analyse Mathematique, 134(2) pp. 801–810.



Let $f$ be a transcendental entire function and let $U$ be a univalent Baker domain of $f$. We prove a new result about the boundary behaviour of conformal maps and use this to show that the non-escaping boundary points of $U$ form a set of harmonic measure zero with respect to $U$. This leads to a new sufficient condition for the escaping set of $f$ to be connected, and also a new general result on Eremenko's conjecture.

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