The dynamics and geometry of the Fatou functions

Abstract

We deal with the Fatou functions $f_\lambda(z)=z+e^{-z}+\lambda$, Re$\lambda\ge1$. We consider the set $J_r(f_\lambda)$ consisting of those pointsof the Julia set of $f_\lambda$ whose real parts do not escape to infinity underpositive iterates of $f_\lambda$. Our ultimate result is that the function$\lambda\mapsto$HD$(J_r(f_\lambda))$ is real-analytic. In order to prove it wedevelop the thermodynamic formalism of potentials of the form$-t$log$|F_\lambda'|$, where $F_\lambda$ is the projection of $f_\lambda$ to theinfinite cylinder. It includes appropriately definedtopological pressure, Perron-Frobenius operators, geometric andinvariant generalized conformal measures (Gibbs states). We show thatour Perron-Frobenius operators are quasicompact, that they embed into afamily of operators depending holomorphically on an appropriate parameterand we obtain severalother properties of these operators. We prove an appropriateversion of Bowen's formula that the Hausdorff dimension of the set$J_r(f_\lambda)$ is equal to the unique zero of the pressurefunction. Since the formulafor the topological pressure is independent of the set $J_r(f_\lambda)$,Bowen's formula also indicates that $J_r(f_\lambda)$ is the right set to dealwith. What concerns geometry of the set $J_r(f_\lambda)$ we also provethat the HD$(J_r(f_\lambda))$-dimensional Hausdorff measure of the set $J_r(F_\lambda)$is positive and finite whereas its HD$(J_r(f_\lambda))$-dimensionalpacking measure is locally infinite. This last property allows us toconclude that HD$(J_r(f_\lambda))<2$.We also study in detail the properties ofquasiconformal conjugations between the maps $f_\lambda$. As a byproduct ofour main courseof reasoning we prove stochastic properties of the dynamical systemgenerated by $F_\lambda$ and the invariant Gibbs states $\mu_t$ such as theCentral Limit Theorem and the exponential decay of correlations.