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- W1987372289 abstract "Ergodic theory originally arose from study of the time evolution of a mechanical system. If T, denotes the transformation of the phase space of a system which takes a state of the system to its state t time units later, then on physical grounds (T,: t E @ obeys the group property that T,,, = T,T, , i.e., forms a flow with parameter group R. Liouville discovered that a certain natural measure on phase space is preserved by each T, , reducing statistical mechanics to the study of asymptotic properties of flows of measure preserving transformations. This approach was simplified by discretizing time and considering only (Tnt,: n E Z] for some fixed to . This has the advantage that only the iterates of a fixed transformation are in question. Starting with the ergodic theorems of Birkhoff and von Neumann in 1931, an interesting theory of measure preserving transformations has been erected. Occasionally the continuous time case was investigated, notably by Weiner [ 171, Ambrose [l], and Ambrose and Kakutani [2]. However, developments centered on measure preserving transformations. In the last decade some progress has been made in classifying measure preserving transformations. Kolmogorov introduced a new numerical invariant in 1958 called entropy which was suggested by Shannon’s work on information theory. This invariant distinguished between some transformations which no one could previously tell apart, and some intriguing conjectures arose. In the past years a series of papers by Ornstein has completely solved a number of major problems, including the isomorphism problem for Bernoulli shifts. A basic technique used by Ornstein in these papers is a theorem of Rokhlin on the periodic approximation of transformations. Some recent work on statistical mechanics by Ruelle [16] and others" @default.
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- W1987372289 date "1975-02-01" @default.
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- W1987372289 title "Locally compact measure preserving flows" @default.
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- W1987372289 doi "https://doi.org/10.1016/0001-8708(75)90133-4" @default.
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