The Dynamics and Geometry of Semi-Hyperbolic Rational Semigroups

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 $tilde mu _tcirc pi _mathcal {U}^{-1}$ 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 $m_tcirc p_2^{-1}$ and $mu _tcirc p_2^{-1}$ versus Hausdorff Measures $mathrm {H}_{t^kappa }$ and $mathrm {H}_{t^kappa exp left (csqrt {log (1/t)log ^3(1/t)}right )}$ 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