Titlebook: Basic ergodic theory; M. G. Nadkarni Book 2013Latest edition Hindustan Book Agency (India) 2013

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Additional Topics,Liouville’s theorem has its origin in classical mechanics. In its simplified version it gives a necessary and sufficient condition for a flow of homeomor-phisms on an open subset in ?. to be volume preserving. Following K. R. Parthasarathy [8] we give this version first, followed by a discussion of its version in classical mechanics.

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,H. Dye’s Theorem, admit Borel cross-sections. We will therefore assume in the rest of this chapter that . and . are free and their orbit spaces do not admit Borel cross-sections. The first important result on orbit equivalence was obtained by H. Dye [2] and the main aim of this chapter is to prove his theorem.

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Bernoulli Shift and Related Concepts,hift and the related concept of .-automorphism at an elementary level. Bernoulli shifts provide us with examples of mixing measure preserving automorphisms. The discussion here follows closely the exposition in Patrick Billingsley [1].

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Discrete Spectrum Theorem,ivalent. Let us say that . and . are spectrally isomorphic if . and . are unitarily equivalent. If . and . are spectrally isomorphic and . is ergodic then . is ergodic, because . is ergodic if and only if 1 is a simple eigenvalue of . hence also of ., which in turn implies the ergodicity of .. Simil