Hausdorff dimension and measures on Julia sets of some meromorphic maps
Barański, Krzysztof
Fundamenta Mathematicae, Tome 146 (1995), p. 239-260 / Harvested from The Polish Digital Mathematics Library
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We study the Julia sets for some periodic meromorphic maps, namely the maps of the form f(z)=h(exp2πiTz) where h is a rational function or, equivalently, the maps ˜f(z)=exp(2πih(z)). When the closure of the forward orbits of all critical and asymptotic values is disjoint from the Julia set, then it is hyperbolic and it is possible to construct the Gibbs states on J(˜f) for -α log |˜˜f|. For ˜α = HD(J(˜f)) this state is equivalent to the ˜α-Hausdorff measure or to the ˜α-packing measure provided ˜α is greater or smaller than 1. From this we obtain some lower bound for HD(J(f)) and real-analyticity of HD(J(f)) with respect to f. As an example the family fλ(z)=λtanz is studied. We estimate HD(J(fλ)) near λ = 0 and show it is a monotone function of real λ.

Publié le : 1995-01-01
EUDML-ID : urn:eudml:doc:212087 
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     title = {Hausdorff dimension and measures on Julia sets of some meromorphic maps},
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     year = {1995},
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Barański, Krzysztof. Hausdorff dimension and measures on Julia sets of some meromorphic maps. Fundamenta Mathematicae, Tome 146 (1995) pp. 239-260. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv147i3p239bwm/

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