Ergodic Theory – Finite and Infinite, Thermodynamic Formalism, Symbolic Dynamics and Distance Expanding Maps
- 458 pages
- English
- PDF
- Available on iOS & Android
Ergodic Theory – Finite and Infinite, Thermodynamic Formalism, Symbolic Dynamics and Distance Expanding Maps
About this book
The book contains a detailed treatment of thermodynamic formalism on general compact metrizable spaces. Topological pressure, topological entropy, variational principle, and equilibrium states are presented in detail. Abstract ergodic theory is also given a significant attention. Ergodic theorems, ergodicity, and Kolmogorov-Sinai metric entropy are fully explored. Furthermore, the book gives the reader an opportunity to find rigorous presentation of thermodynamic formalism for distance expanding maps and, in particular, subshifts of finite type over a finite alphabet. It also provides a fairly complete treatment of subshifts of finite type over a countable alphabet. Transfer operators, Gibbs states and equilibrium states are, in this context, introduced and dealt with. Their relations are explored. All of this is applied to fractal geometry centered around various versions of Bowen's formula in the context of expanding conformal repellors, limit sets of conformal iterated function systems and conformal graph directed Markov systems. A unique introduction to iteration of rational functions is given with emphasize on various phenomena caused by rationally indifferent periodic points. Also, a fairly full account of the classicaltheory of Shub's expanding endomorphisms is given; it does not have a book presentation in English language mathematical literature.
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Information
Table of contents
- Preface
- List of Figures
- Introduction to Volume 1
- Contents
- 1 Dynamical systems
- 2 Homeomorphisms of the circle
- 3 Symbolic dynamics
- 4 Distance expanding maps
- 5 (Positively) expansive maps
- 6 Shub expanding endomorphisms
- 7 Topological entropy
- 8 Ergodic theory
- 9 Measure-theoretic entropy
- 10 Infinite invariant measures
- 11 Topological pressure
- 12 The variational principle and equilibrium states
- Appendix A. A selection of classical results
- Bibliography
- Index