Abstract
In the previous chapter we have introduced the notion of a measure-preserving system and seen a number of examples. We now begin with their systematic study. In particular we shall define invertibility of a measure-preserving system, prove the classical recurrence theorem of Poincaré, and introduce the central notion of an ergodic system.
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- 1.
Behold, we know what thou teachest: that all things eternally return, and ourselves with them, and that we have already existed times without number, and all things with us.
- 2.
Also sprach Zarathustra, Teil III, Der Genesende. From: Werke II, hrsg. v. Karl Schlechta, Darmstadt, 1997 ⋅ Translation from: Thus Spake Zarathustra, Part III, The Convalescent, translated by Thomas Common, Wilder Publications, 2008.
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© 2015 Tanja Eisner, Bálint Farkas, Markus Haase, and Rainer Nagel
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Eisner, T., Farkas, B., Haase, M., Nagel, R. (2015). Recurrence and Ergodicity. In: Operator Theoretic Aspects of Ergodic Theory. Graduate Texts in Mathematics, vol 272. Springer, Cham. https://doi.org/10.1007/978-3-319-16898-2_6
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DOI: https://doi.org/10.1007/978-3-319-16898-2_6
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