The Memoirs of the AMS is devoted to the publication of new research in all areas of pure and applied mathematics. The Memoirs is designed particularly to publish long papers of groups of cognate papers in book form, and is under the supervision of the Editorial Committee of the AMS journal Transactions of the American Mathematical Society. All papers are peer-reviewed.
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
Jason Atnip, University of New South Wales, Sydney, Australia.
Hiroki Sumi, Kyoto University, Japan.
Mariusz Urbanski, University of North Texas, Denton, Texas.
