Dynamics and Geometry of Semi-Hyperbolic Rational Semigroups

Jason Atnip, University of New South Wales, Sydney, Australia.Hiroki Sumi, Kyoto University, Japan.Mariusz Urbanski, University of North Texas, Denton, Texas.

• Chapters1. Introduction2. General Preliminaries on Rational Semigroups1. Ergodic Theory and Dynamics of Finitely Generated *Semi-Hyperbolic Rational Semigroups3. Basic Properties of Semi-hyperbolic and *Semi-Hyperbolic Rational Semigroups4. The Conformal and Invariant Measures $m_t$ and $\mu _t$ for $\tilde f:J(\tilde f)\longrightarrow J(\tilde f)$2. Ergodic Theory and Dynamics of Totally and Finely Non-Recurrent Rational Semigroups5. Totally Non-Recurrent and Finely Non-Recurrent Rational Semigroups6. Nice Sets (Families)7. The Behavior of the Absolutely Continuous Invariant Measures $\mu _t$ Near Critical Points8. Small Pressure $\mathrm {P}_V^\Xi (t)$9. Symbol Space Thermodynamic Formalism Associated to Nice Families: Real Analyticity of the Original Pressure $\mathrm {P}(t)$10. Invariant Measures: $\mu _t$ versus $\tilde \mu _t\circ \pi _\mathcal {U}^{-1}$Finiteness of $\mu _t$11. Variational Principle: The Invariant Measures $\mu _t$ are the Unique Equilibrium States12. Decay of Correlations, Central Limit Theorems, the Law of Iterated Logarithm: The Method of Lai-Sang Young Towers3. Geometry of Finely Non-Recurrent Rational Semigroups Satisfying the Nice Open Set Condition13. Nice Open Set Condition (for any Rational Semigroup)14. Hausdorff Dimension of Invariant Measures $\mu _t$ and Multifractal Analysis of Lyapunov Exponents15. Measures $m_t\circ p_2^{-1}$ and $\mu _t\circ p_2^{-1}$ versus Hausdorff Measures $\mathrm {H}_{t^\kappa }$ and $\mathrm {H}_{t^\kappa \exp \left (c\sqrt {\log (1/t)\log ^3(1/t)}\right )}$16. $\operatorname {HD}(J(G))$ versus Hausdorff Dimension of Fiber Julia Sets $J_\omega $, $\omega \in \Sigma _u$17. Examples4. AppendicesA. Absolutely Continuous $\sigma $-finite Invariant Measures: Martens MethodB. Corrected Proofs of Lemma 7.9 and Lemma 7.10 from \cite{SUSH}C. Definitions of Classes of Rational Semigroups Used and Relations Between ThemD. Open Problems