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arXiv · 1511.02897

Escaping points in the boundaries of Baker domains

Krzysztof Barański·Núria Fagella·Xavier Jarque·Bogusława Karpińska
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Abstract

We study the dynamical behaviour of points in the boundaries of simply connected invariant Baker domains $U$ of meromorphic maps $f$ with a finite degree on $U$. We prove that if $f|_U$ is of hyperbolic or simply parabolic type, then almost every point in the boundary of $U$ with respect to harmonic measure escapes to infinity under iteration. On the contrary, if $f|_U$ is of doubly parabolic type, then almost every point in the boundary of $U$ with respect to harmonic measure has dense forward trajectory in the boundary of $U$, in particular the set of escaping points in the boundary of $U$ has harmonic measure zero. We also present some extensions of the results to the case when $f$ has infinite degree on $U$, including classical Fatou example.

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Krzysztof Barański, Núria Fagella, Xavier Jarque, Bogusława Karpińska. 2015-11-09. Escaping points in the boundaries of Baker domains. https://doi.org/10.1007/s11854-019-0011-0

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