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Hausdorff dimension of hairs and ends for entire maps of finite order

Published online by Cambridge University Press:  01 November 2008

KRZYSZTOF BARAŃSKI
KRZYSZTOF BARAŃSKI*
Affiliation:
Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland. e-mail: baranski@mimuw.edu.pl
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Abstract

We study transcendental entire maps f of finite order, such that all the singularities of f−1 are contained in a compact subset of the immediate basin B of an attracting fixed point of f. Then the Julia set of f consists of disjoint curves tending to infinity (hairs), attached to the unique point accessible from B (endpoint of the hair). We prove that the Hausdorff dimension of the set of endpoints of the hairs is equal to 2, while the union of the hairs without endpoints has Hausdorff dimension 1, which generalizes the result for exponential maps. Moreover, we show that for every transcendental entire map of finite order from class (i.e. with bounded set of singularities) the Hausdorff dimension of the Julia set is equal to 2.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 145 , Issue 3 , November 2008 , pp. 719 - 737
Copyright
Copyright © Cambridge Philosophical Society 2008

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