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

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

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