The 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.

Chapters 1. Introduction 2. General Preliminaries on Rational Semigroups 1. Ergodic Theory and Dynamics of Finitely Generated *Semi-Hyperbolic Rational Semigroups 3. Basic Properties of Semi-hyperbolic and *Semi-Hyperbolic Rational Semigroups 4. 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 Semigroups 5. Totally Non-Recurrent and Finely Non-Recurrent Rational Semigroups 6. Nice Sets (Families) 7. The Behavior of the Absolutely Continuous Invariant Measures $\mu _t$ Near Critical Points 8. 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 States 12. Decay of Correlations, Central Limit Theorems, the Law of Iterated Logarithm: The Method of Lai-Sang Young Towers 3. Geometry of Finely Non-Recurrent Rational Semigroups Satisfying the Nice Open Set Condition 13. Nice Open Set Condition (for any Rational Semigroup) 14. Hausdorff Dimension of Invariant Measures $\mu _t$ and Multifractal Analysis of Lyapunov Exponents 15. 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. Examples 4. Appendices A. Absolutely Continuous $\sigma $-finite Invariant Measures: Martens Method B. Corrected Proofs of Lemma 7.9 and Lemma 7.10 from \cite{SUSH} C. Definitions of Classes of Rational Semigroups Used and Relations Between Them D. Open Problems