Real Analyticity of Hausdorff Dimension of Finer Julia Sets of Exponential Family

We deal with all the mappings f (z) = e that have an attracting periodic orbit. We consider the set J r (f ) consisting of those points of the Julia set of f that do not escape to in nity under positive iterates of f . Our ultimate result is that the function 7! HD(J r (f )) is real analytic. In order to prove it we develop the thermodynamic formalism of potentials of the form t log jF j, where F is the natural map associated with f closely related to the corresponding map introduced in [UZd]. It includes appropriately de ned topological pressure, Perron-Frobenius operators, geometric and invariant generalized conformal measures (Gibbs states). We show that our Perron-Frobenius operators are quasicompact, that they embed into a family of operators depending holomorphically on an appropriate parameter and we obtain several other properties of these operators. We also study in detail the properties of quasiconformal conjugacies between the maps f . As a byproduct of our main course of reasoning we prove stochastic properties of the dynamical system generated by F and the invariant Gibbs states t such as the Central Limit Theorem and the exponential decay of correlations