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Memoirs of the American Mathematical Society
The Dynamics and Geometry of Semi-Hyperbolic Rational Semigroups

The Dynamics and Geometry of Semi-Hyperbolic Rational Semigroups

by Jason Atnip, Hiroki Sumi and Mariusz Urbanski

Published by: American Mathematical Society

Series: Memoirs of the American Mathematical Society

Imprint: American Mathematical Society

192 Pages, 178.00 × 254.00 mm

Paperback
9781470473082
Published: June 2025

£73.00

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Description
Contents
Authors

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