Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

The dynamics and geometry of the Fatou functions

Pages: 291 - 338, Volume 13, Issue 2, July 2005      doi:10.3934/dcds.2005.13.291

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Janina Kotus - Faculty of Mathematics & Information Sciences, Warsaw University of Technology, Plac Politechnki 1, Warsaw 00-661, Poland (email)
Mariusz UrbaƄski - Department of Mathematics, University of North Texas, P.O. Box 311430, Denton, TX 76203-1430, United States (email)

Abstract: We deal with the Fatou functions $f_\lambda(z)=z+e^{-z}+\lambda$, Re$\lambda\ge 1$. We consider the set $J_r(f_\lambda)$ consisting of those points of the Julia set of $f_\lambda$ whose real parts do not escape to infinity under positive 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 we develop the thermodynamic formalism of potentials of the form $-t$log$|F_\lambda'|$, where $F_\lambda$ is the projection of $f_\lambda$ to the infinite cylinder. It includes appropriately defined 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 prove an appropriate version of Bowen's formula that the Hausdorff dimension of the set $J_r(f_\lambda)$ is equal to the unique zero of the pressure function. Since the formula for 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 deal with. What concerns geometry of the set $J_r(f_\lambda)$ we also prove that the HD$(J_r(f_\lambda))$-dimensional Hausdorff measure of the set $J_r(F_\lambda)$ is positive and finite whereas its HD$(J_r(f_\lambda))$-dimensional packing measure is locally infinite. This last property allows us to conclude that HD$(J_r(f_\lambda))<2$. We also study in detail the properties of quasiconformal conjugations between the maps $f_\lambda$. As a byproduct of our main course of reasoning we prove stochastic properties of the dynamical system generated by $F_\lambda$ and the invariant Gibbs states $\mu_t$ such as the Central Limit Theorem and the exponential decay of correlations.

Keywords:  Fatou functions, Julia set, Hausdorff dimension, topological pressure, Perron-Frobenius operator, conformal measures, Hausdorff and packing measures, real analyticity of Hausdorff dimension.
Mathematics Subject Classification:  Primary: 37F35; Secondary: 37F10, 30D05.

Received: May 2004;      Revised: March 2005;      Available Online: April 2005.