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OverviewThe 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. Full Product DetailsAuthor: Jason Atnip , Hiroki Sumi , Mariusz UrbanskiPublisher: American Mathematical Society Imprint: American Mathematical Society ISBN: 9781470473082ISBN 10: 1470473089 Pages: 192 Publication Date: 30 June 2025 Audience: Professional and scholarly , Professional & Vocational Format: Paperback Publisher's Status: Active Availability: Not yet available ![]() This item is yet to be released. You can pre-order this item and we will dispatch it to you upon its release. Table of ContentsChapters 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 ProblemsReviewsAuthor InformationJason Atnip, University of New South Wales, Sydney, Australia. Hiroki Sumi, Kyoto University, Japan. Mariusz Urbanski, University of North Texas, Denton, Texas. Tab Content 6Author Website:Countries AvailableAll regions |