Hausdorff dimension and measures on Julia sets of some meromorphic maps

• Autorzy: Krzysztof Barański
• Języki publikacji: EN

Abstrakty

EN

We study the Julia sets for some periodic meromorphic maps, namely the maps of the form $f(z) = h(exp \frac{2πi}{T}z)$ where h is a rational function or, equivalently, the maps $˜f(z) = exp (\frac{2πi}{h}(z))$. 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 $f_λ(z)=λ tan z$ is studied. We estimate $HD(J(f_λ))$ near λ = 0 and show it is a monotone function of real λ.

Kategorie tematyczne

• 28A78: Hausdorff and packing measures

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 1995
• Tom: 147
• Numer: 3
• Strony: 239-260

Daty

wydano

1995

otrzymano

1994-06-14

Twórcy

autor

Krzysztof Barański

• Institute of Mathematics, Warsaw University, Banacha 2, 02-097 Warszawa, Poland

Bibliografia

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