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

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,Name / Herkunft / Lebensumstände / Bildung,We have seen that a measure preserving automorphism . on a probability space (., ., .) is ergodic if and only if for all ., . ∈ .,.. Two properties stronger than ergodieity discovered by Koopman and von Neumann [2] will now be discussed.

PAD416

capsaicin

Henry E. Kyburg, Jr. & Isaac LeviLet (., .) be a standard Borel space. A group ., . ∈ ℝ, of Borel automorphisms on (., .) is called a jointly measurable flow, or simply a flow, if

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https://doi.org/10.1007/978-94-009-7718-1Liouville’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 ℝ. 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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,The Poincaré Recurrence Lemma,Let . be a non-empty set. A .-algebra . on . is a non-empty collection of subsets of . which is closed under countable unions and complements. A set together with a .-algebra . is called a Borel space or a Borel structure (., .).

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抱狗不敢前

Ergodicity,A measure preserving Borel automorphism . on a probability space (., ., .) is said to be ergodic if for every . ∈ . invariant under ., .(.) = 0 or .(. − .) = 0.

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Mixing Conditions and Their Characterisations,We have seen that a measure preserving automorphism . on a probability space (., ., .) is ergodic if and only if for all ., . ∈ .,.. Two properties stronger than ergodieity discovered by Koopman and von Neumann [2] will now be discussed.

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Flows and Their Representations,Let (., .) be a standard Borel space. A group ., . ∈ ℝ, of Borel automorphisms on (., .) is called a jointly measurable flow, or simply a flow, if