Abstract
The ergodic theorem was proved by G.D. Birkhoff in 1932. At the time, there were already examples of ergodic systems. They came from probability theory and model random phenomena like throwing a die or drawing balls from an urn. It is therefore not surprising to see ergodicity appear in this context.
The author has had complaints about too much detail missing in the presentation of the material in the latter paper. This has been rectified in the present paper. E. Hopf (1902–1983)
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References
Arnol′d, V.I.: Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics, vol. 60. Springer, New York (1978)
Beardon, A.F.: The Geometry of Discrete Groups. Graduate Texts in Mathematics, vol. 91. Springer, New York (1983)
Gallot, S., Hulin, D., Lafontaine, J.: Riemannian Geometry, 3rd edn. Universitext. Springer, Berlin (2004)
Katok, S.: Fuchsian Groups. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL (1992)
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Coudène, Y. (2016). The Hopf Argument. In: Ergodic Theory and Dynamical Systems. Universitext. Springer, London. https://doi.org/10.1007/978-1-4471-7287-1_4
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DOI: https://doi.org/10.1007/978-1-4471-7287-1_4
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