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The existence of conformal measures for some transcendental meromorphic functions. (English) Zbl 1207.37031

Devaney, Robert L. (ed.) et al., Complex dynamics. Twenty-five years after the appearance of the Mandelbrot set. Proceedings of an AMS-IMS-SIAM joint summer research conference, Snowbird, UT, USA, June 13–17, 2004. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3625-0/pbk). Contemporary Mathematics 396, 169-201 (2006).
Summary: We study the dynamics of transcendental meromorphic functions of the form \(f(z)= R(e^z)\) where \(R\) is non-constant rational map and \(f\) has an asymptotic value \(\xi\) such that \(f^q(\xi)=\infty\) for some integer \(q\geq 0\). We investigate the existence of an \(h\)-conformal measure with \(1<h<2\), that is conformal for a projection \(F\) of the map \(f\) on the cylinder \(\mathbb C/_{2\pi i\mathbb Z}\). This gives us an estimate of the Hausdorff dimension of the radial Julia set \(J^r_f\). This also shows that the hyperbolic dimension is strictly less than the Hausdorff dimension of \(J_f\). At the end we study the Hausdorff dimension and the Lebesgue measure of the entire Julia set and its hyperbolic subsets. We in particular give an example of a non-entire meromorphic function with fat Julia set.
For the entire collection see [Zbl 1084.37002].

MSC:

37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
37D35 Thermodynamic formalism, variational principles, equilibrium states for dynamical systems
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
37F35 Conformal densities and Hausdorff dimension for holomorphic dynamical systems
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