Hausdorff measure of hairs without endpoints in the exponential family
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Devaney and Krych showed that for \(0<\lambda <1/e\) the Julia set of \(\lambda e^z\) 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.
KeywordsIteration Fatou set Julia set Hair Endpoint Hausdorff dimension Hausdorff measure
Mathematics Subject Classification37F10 30D05
We thank the referee for valuable comments. The second author was supported by the scholarship from China Scholarship Council (No. 201206105015), and would express her thanks for the hospitality of Mathematisches Seminar at Christian-Albrechts-Universität zu Kiel.
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