Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

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

KRZYSZTOF BARAŃSKIa1

a1 Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland. e-mail: baranski@mimuw.edu.pl

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.

(Received April 24 2006)

(Revised November 27 2007)

Footnotes

† Research supported by Polish MNiSW Grant N N201 0234 33 and EU FP6 Marie Curie ToK Programme SPADE2.