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On notions of determinism in topological dynamics

  • MICHAEL HOCHMAN (a1)

Abstract

We examine the relations between topological entropy, invertibility, and prediction in topological dynamics. We show that topological determinism in the sense of Kamińsky, Siemaszko, and Szymański imposes no restriction on invariant measures except zero entropy. Also, we develop a new method for relating topological determinism and zero entropy, and apply it to obtain a multidimensional analog of this theory. We examine prediction in symbolic dynamics and show that while the condition that each past admits a unique future only occurs in finite systems, the condition that each past has a bounded number of futures imposes no restriction on invariant measures except zero entropy. Finally, we give a negative answer to a question of Eli Glasner by constructing a zero-entropy system with a globally supported ergodic measure in which every point has multiple preimages.

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[1]Bobok, J.. The topological entropy versus level sets for interval maps. Studia Math. 152(3) (2002), 249261.
[2]Bobok, J. and Nitecki, Z.. Topological entropy of m-fold maps. Ergod. Th. & Dynam. Sys. 25(2) (2005), 375401.
[3]Cheng, W.-C. and Newhouse, S. E.. Pre-image entropy. Ergod. Th. & Dynam. Sys. 25(4) (2005), 10911113.
[4]Eigen, S. J. and Prasad, V. S.. Multiple Rokhlin tower theorem: a simple proof. New York J. Math. 3A (1997–1998), 1114 (Proceedings of the New York Journal of Mathematics Conference, 9–13 June 1997) (electronic).
[5]Fiebig, D., Fiebig, U.-R. and Nitecki, Z. H.. Entropy and preimage sets. Ergod. Th. & Dynam. Sys. 23(6) (2003), 17851806.
[6]Kamiński, B., Siemaszko, A. and Szymański, J.. The determinism and the Kolmogorov property in topological dynamics. Bull. Pol. Acad. Sci. Math. 51(4) (2003), 401417.
[7]Kamiński, B., Siemaszko, A. and Szymański, J.. Extreme relations for topological flows. Bull. Pol. Acad. Sci. Math. 53(1) (2005), 1724.
[8]Lind, D. and Marcus, B.. An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge, 1995.
[9]Nitecki, Z. and Przytycki, F.. Preimage entropy for mappings. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 9(9) (1999), 18151843; Discrete dynamical systems.
[10]Shields, P.. The Theory of Bernoulli Shifts (Chicago Lectures in Mathematics). The University of Chicago Press, Chicago–London, 1973.
[11]Walters, P.. An Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79). Springer, New York, 1982.
[12]Weiss, B.. Multiple recurrence and doubly minimal systems. Topological Dynamics and Applications (Minneapolis, MN, 1995) (Contemporary Mathematics, 215). American Mathematical Society, Providence, RI, 1998, pp. 189196.

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On notions of determinism in topological dynamics

  • MICHAEL HOCHMAN (a1)

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