One of the central themes in Dynamical Systems and Ergodic Theory is that of recurrence, which is a circle of results concerning how points in measurable dynamical systems return close to themselves under iteration. There are several types of recurrent behavior (exact recurrence, Poincaré recurrence, coherent recurrence , ...) for some classes of measurability-preserving discrete time dynamical systems. P. Johnson and A. Sklar in [Recurrence and dispersion under iteration of Čebyšev polynomials. J. Math. Anal. Appl. 54 (1976), no. 3, 752-771] regard the third type („ coherent recurrence” for measurability-preserving transformations) as being of at least equal physical significance, and this type of recurrence fails for Čebyšev polynomials. They also found that there is considerable evidence to support a conjecture that no (strongly) mixing transformation can exhibit coherent recurrence. (This conjecture has been proved by R. E. Rice in [On mixing transformations. Aequationes Math. 17 (1978), no. 1, 104-108].)
Arnold's cat map (considered as chaotic) is strongly mixing transformation (of the Torus ) and so ergodic transformation. However, R. E. Rice in [Aequationes Math. 17 (1978), no. 1, 104-108] proved that no (strongly) mixing transformation can exhibit coherent recurrence.
Arnold's cat map (considered as chaotic) is strongly mixing transformation (of the Torus ) and so ergodic transformation. However, R. E. Rice in [Aequationes Math. 17 (1978), no. 1, 104-108] proved that no (strongly) mixing transformation can exhibit coherent recurrence.
For “the definition of coherent recurrence (for measure/ measurability-preserving transformations) ” see, e.g., in:
1) [P. Johnson and A. Sklar, J. Math. Anal. Appl. 54 (1976), no. 3, 752-771],
2) [R. E. Rice, Aequationes Math. 17 (1978), no. 1, 104-108],
3) H. Fatkić, “O vjerovatnosnim metričkim prostorima i ergodičnim transformacijama (with a summary in English)” (https://www.researchgate.net/publication/275893244);
4) [ B. Schweizer, A. Sklar, Probabilistic metric spaces, North-Holland Ser. Probab. Appl. Math., North-Holland, New York, 1983; second edition, Dover, Mineola, NY, 2005 (Definition 11.4.3, p.185)].
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