Finer fractal geometry for analytic families of conformal dynamical systems

Abstract

We prove several results establishing real analyticity of Hausdorff dimensions of limit sets of analytic families of conformal graph directed Markov systems. With this tool and with iterated function systems resulting from the existence of nice sets in the sense of Rivera-Letelier, we prove that the canonical Hausdorff measure restricted to the radial Julia set of a tame meromorphic function (can be rational) is σ-finite and that the Hausdorff dimension of the radial Julia sets for fairly general families of meromorphic functions (can be rational) is real analytic.