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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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Topological Aspects of Dynamics of Pairs, Tuples and Sets
TL;DR: In this article, an exposition on local aspects of dynamics of pairs, or more generally tuples and sets, is presented, motivated by a strong belief of the authors that very often global properties of dynamics can be deduced from its local behavior.
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Probabilistic Averages of Jacobi Operators
TL;DR: In this article, the Lyapunov exponent and the integrated density of states for general Jacobi operators are studied and the main result is that questions about these can be reduced to questions about ergodic Jacobi operator.
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Topological entropy of continuous functions on topological spaces
Lei Liu,Yangeng Wang,Guo Wei +2 more
TL;DR: A new definition of topological entropy for continuous mappings on arbitrary topological spaces (compactness, metrizability, even axioms of separation not necessarily required) is proposed and investigated, and fundamental properties of the new entropy are investigated and compared with the existing ones.
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Dominated splitting and Pesin's entropy formula
Wenxiang Sun,Xueting Tian +1 more
TL;DR: In this article, the authors obtained that Pesin's entropy formula always holds for (1) volume-preserving Anosov diffeomorphisms, (2) volumepreserving partially hyperbolic diffEomorphisms with one-dimensional center bundle, (3) volume preserving diffEmorphisms far away from homoclinic tangency, and (4) generic manifold diffemeomorphisms.
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Geometry of self-similar measures
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