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Mixing Conditions and Their Characterisations

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Part of the Texts and Readings in Mathematics book series (volume 6)

Abstract

We have seen that a measure preserving automorphism σ on a probability space (X, Ɓ, m) is ergodic if and only if for all A, BƁ,
$$\frac{1}{n}\sum\limits_{k = 0}^{n - 1} {m\left( {A \cap {\sigma ^k}B} \right) \to m\left( A \right)\;\cdot\;m\left( B \right)} \;as\;n \to \infty$$
. Two properties stronger than ergodieity discovered by Koopman and von Neumann [2] will now be discussed.

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Bibliography

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    S. Kalikow. Two Fold Mixing Implies Three Fold Mixing for Rank One Transformations, Ergod. Th. and Dynam. Sys. 4 (1984), 237–259.CrossRefzbMATHGoogle Scholar
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    B. O. Koopman and J. von Neumann. Dynamical Systems of Continuous Spectra. Proc. Nat. Acad. Sci. (U. S. A.), 18 (1932), 255–263. John von Neumann: Collected Works, vol. II. Pergamon Press, 278–286.CrossRefzbMATHGoogle Scholar
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    K. Petersen. Ergodic Theory, Cambridge University Press, 1983.CrossRefzbMATHGoogle Scholar
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    V. A. Rokhlin. On The Endomorphisms of Compact Commutative Groups, Izv. Acad. Sci. USSR, Ser Mat 13 (1949),, 323–340.MathSciNetGoogle Scholar
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    P. Walters. Ergodic Theory: Introductory Lectures, Springer-Verlag, 1975.CrossRefzbMATHGoogle Scholar

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© Hindustan Book Agency 2013

Authors and Affiliations

  1. 1.University of MumbaiIndia

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