Hausdorff dimension and measures on Julia sets of some meromorphic maps
Volume 147 / 1995
Abstract
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 $J(\widetilde f)$ for $-α \log |\widetilde f'|$. For $\widetilde α ={\rm HD}(J(\widetilde f))$ this state is equivalent to the $\widetilde α$-Hausdorff measure or to the $\widetilde α$-packing measure provided $\widetilde α$ is greater or smaller than 1. From this we obtain some lower bound for ${\rm HD}(J(f))$ and real-analyticity of ${\rm HD}(J(f))$ with respect to $f$. As an example the family $f_λ(z)=λ \operatorname{tan} z$ is studied. We estimate ${\rm HD}(J(f_λ))$ near $λ = 0$ and show it is a monotone function of real λ.
