遍历理论
遍历的定义
考虑适定的函数f的时间平均。这定义为从某个初始点x开始的时间间隔T的取值的平均。
再考虑f的空间平均和相位平均,定义为
其中μ是概率空间的测度。
一般来说,时间平均和空间平均可能不同。 但是若变换是遍历的,而该测度不变,则时间均值和空间均值几乎处处相等。这就是著名的遍历定理,其抽象形式由乔治·戴维·伯克霍夫给出。平均分布定理是遍历定理的一个特殊情况,专门处理单位间隔上的概率分布。
参看
- 始态复现定理
历史参考
- G. D. Birkhoff, Proof of the ergodic theorem, (1931), Proceedings of the National Academy of Sciences USA, 17 pp 656-660.
- E. Hopf, Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung, (1939) Leipzig Ber. Verhandl. Sächs. Akad. Wiss. 91, p.261-304.
- S. V. Fomin and I. M. Gelfand, Geodesic flows on manifolds of constant negative curvature, (1952) Uspehi Mat. Nauk 7 no. 1. p. 118-137.
- F. I. Mautner, Geodesic flows on symmetric Riemann spaces, (1957) Ann. of Math. 65 p. 416-431.
- C. C. Moore, Ergodicity of flows on homogeneous spaces, (1966) Amer. J. Math. 88, p.154-178.
现代参考
- Vladimir Igorevich Arnol'd and André Avez, Ergodic Problems of Classical Mechanics. New York: W.A. Benjamin. 1968.
- Leo Breiman, Probability. Original edition published by Addison-Wesley, 1968; reprinted by Society for Industrial and Applied Mathematics, 1992. ISBN 978-0-89871-296-4. (See Chapter 6.)
- Peter Walters, An introduction to ergodic theory, Springer, New York, 1982, ISBN 978-0-387-95152-2.
- Tim Bedford, Michael Keane and Caroline Series, eds.. . Oxford University Press. 1991. ISBN 978-0-19-853390-0. (A survey of topics in ergodic theory; with exercises.)
- Joseph M. Rosenblatt and Máté Weirdl, Pointwise ergodic theorems via harmonic analysis, (1993) appearing in Ergodic Theory and its Connections with Harmonic Analysis, Proceedings of the 1993 Alexandria Conference, (1995) Karl E. Petersen and Ibrahim A. Salama, eds., Cambridge University Press, Cambridge, ISBN 978-0-521-45999-0. (An extensive survey of the ergodic properties of generalizations of the equidistribution theorem of shift maps on the unit interval. Focuses on methods developed by Bourgain.)
関連書籍
- 『エルゴード理論とフラクタル』 釜江哲郎・高橋智 共著 (1993, シュプリンガー・フェアラーク東京, ISBN 4-431-70645-3)
- Probability : Theory and Examples (Richard Durrett, Thomson, ISBN 0-534-42441-4)
- Peter Walters, An Introduction to Ergodic Theory
外部链接
- What Is Ergodicity? An intuitive description of the concept
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