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Escaping points in the boundaries of Baker domains

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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 ofU,with respect to harmonicmeasure, escapes to infinity under iteration of f. 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 the classical Fatou example.

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Correspondence to Ńuria Fagella.

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Research supported by the Polish NCN grant decision DEC-2012/06/M/ST1/00168.

The second and third authors were partially supported by the Spanish grants MTM2014-52209-C2-2-P, MTM2017-86795-C3-3-P, the Maria de Maeztu Excellence Grant MDM-2014-0445 of the BGSMath and the catalan grant 2017SGR1374.

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Barański, K., Fagella, Ń., Jarque, X. et al. Escaping points in the boundaries of Baker domains. JAMA 137, 679–706 (2019). https://doi.org/10.1007/s11854-019-0011-0

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  • DOI: https://doi.org/10.1007/s11854-019-0011-0

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