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arXiv · 1502.01961

Hausdorff measure of hairs without endpoints in the exponential family

Walter Bergweiler·Jun Wang
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Abstract

Devaney and Krych showed that for $0<λ<1/e$ the Julia set of $λe^z$ consists of pairwise disjoint curves, called hairs, which connect finite points, called the endpoints of the hairs, with $\infty$. McMullen showed that the Julia set has Hausdorff dimension $2$ and Karpińska showed that the set of hairs without endpoints has Hausdorff dimension $1$. We study for which gauge functions the Hausdorff measure of the set of hairs without endpoints is finite.

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Walter Bergweiler, Jun Wang. 2015-02-06. Hausdorff measure of hairs without endpoints in the exponential family. https://doi.org/10.1007/s00209-015-1514-8

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