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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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Measure-Preserving Transformations, Copulæ and Compatibility
TL;DR: In order to construct a 3-copula from two given 2-copulas A and B, the *-operation introduced in Darsow et al. is modified, it is shown that A * B is always compatible with A and A, and the set $$\mathcal {D}(A,B)$$ of all copulas compatible with B is studied.
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On Dynamic Topological and Metric Logics
TL;DR: It is shown that for various classes of topological spaces the resulting logics are not Recursively enumerable (and so not recursively axiomatisable) and this gives a ‘negative’ solution to a conjecture of Kremer and Mints.
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Dimension and Measures for a Curvilinear Sierpinski Gasket or Apollonian Packing
TL;DR: In this paper, it was shown that the residual set of a standard Apollonian packing or a curvilinear Sierpinski gasket has a Hausdorff dimension greater than 1.
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A direct proof of the tail variational principle and its extension to maps
TL;DR: In this article, Downarowicz et al. gave an elementary proof of the variational principle for the tail entropy for continuous dynamical systems of a compact metric space and extended the result to the non-invertible case.
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On weakly almost periodic measures
Daniel Lenz,Nicolae Strungaru +1 more
TL;DR: In this article, it was shown that the dynamical hull of a weakly almost periodic measure is a dynamical system with unique minimal component given by the hull of the strongly almost periodic component of the measure.