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An Introduction to Ergodic Theory
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The first part of the text as discussed by the authors provides an introduction to ergodic theory suitable for readers knowing basic measure theory, including recurrence properties, mixing properties, the Birkhoff Ergodic theorem, isomorphism, and entropy theory.Abstract:
This text provides an introduction to ergodic theory suitable for readers knowing basic measure theory. The mathematical prerequisites are summarized in Chapter 0. It is hoped the reader will be ready to tackle research papers after reading the book. The first part of the text is concerned with measure-preserving transformations of probability spaces; recurrence properties, mixing properties, the Birkhoff ergodic theorem, isomorphism and spectral isomorphism, and entropy theory are discussed. Some examples are described and are studied in detail when new properties are presented. The second part of the text focuses on the ergodic theory of continuous transformations of compact metrizable spaces. The family of invariant probability measures for such a transformation is studied and related to properties of the transformation such as topological traitivity, minimality, the size of the non-wandering set, and existence of periodic points. Topological entropy is introduced and related to measure-theoretic entropy. Topological pressure and equilibrium states are discussed, and a proof is given of the variational principle that relates pressure to measure-theoretic entropies. Several examples are studied in detail. The final chapter outlines significant results and some applications of ergodic theory to other branches of mathematics.read more
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Induced topological pressure for countable state Markov shifts
TL;DR: In this paper, the authors introduce the notion of induced topological pressure for countable state Markov shifts with respect to a non-negative scaling function and an arbitrary subset of finite words.
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Interactions, Specifications, DLR probabilities and the Ruelle Operator in the One-Dimensional Lattice
Leandro Cioletti,Artur O. Lopes +1 more
TL;DR: In this paper, the authors describe several different meanings for the concept of Gibbs measure on the lattice $\mathbb{N}$ in the context of finite alphabets (or state space).
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Lyapunov’s direct method in estimates of topological entropy
TL;DR: In this paper, an upper estimate for the topological entropy of a dynamical system defined by a system of ODEs is obtained, which involves the Lyapunov functions and Losinskii's logarithmic norm.
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On topological entropy of billiard tables with small inner scatterers
TL;DR: In this article, the authors studied the topological entropy of a class of billiard tables with strictly convex inner scatterers and showed that the first return map can be made arbitrarily large under the assumption that the inner sparsification is sufficiently small.
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Schrödinger’s cat
TL;DR: In this paper, it was shown that the classical framework of probability spaces, which does not admit a model-theoretical treatment, is equivalent to that of probability algebras which does.