Hostname: page-component-7b7567c8c-kwt9d Total loading time: 0 Render date: 2026-11-14T21:48:11.928Z Has data issue: false hasContentIssue false
  • English
  • Français

On notions of determinism in topological dynamics

Published online by Cambridge University Press:  06 September 2011

MICHAEL HOCHMAN
MICHAEL HOCHMAN*
Affiliation:
Mathematics Department, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544, USA (email: hochman@math.princeton.edu)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

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.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 1 , February 2012, pp. 119 - 140
Copyright
Copyright © Cambridge University Press 2011

References

[1]Bobok, J.. The topological entropy versus level sets for interval maps. Studia Math. 152(3) (2002), 249–261.CrossRefGoogle Scholar
[2]Bobok, J. and Nitecki, Z.. Topological entropy of m-fold maps. Ergod. Th. & Dynam. Sys. 25(2) (2005), 375–401.CrossRefGoogle Scholar
[3]Cheng, W.-C. and Newhouse, S. E.. Pre-image entropy. Ergod. Th. & Dynam. Sys. 25(4) (2005), 1091–1113.CrossRefGoogle Scholar
[4]Eigen, S. J. and Prasad, V. S.. Multiple Rokhlin tower theorem: a simple proof. New York J. Math. 3A (1997–1998), 11–14 (Proceedings of the New York Journal of Mathematics Conference, 9–13 June 1997) (electronic).Google Scholar
[5]Fiebig, D., Fiebig, U.-R. and Nitecki, Z. H.. Entropy and preimage sets. Ergod. Th. & Dynam. Sys. 23(6) (2003), 1785–1806.CrossRefGoogle Scholar
[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), 401–417.Google Scholar
[7]Kamiński, B., Siemaszko, A. and Szymański, J.. Extreme relations for topological flows. Bull. Pol. Acad. Sci. Math. 53(1) (2005), 17–24.CrossRefGoogle Scholar
[8]Lind, D. and Marcus, B.. An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar
[9]Nitecki, Z. and Przytycki, F.. Preimage entropy for mappings. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 9(9) (1999), 1815–1843; Discrete dynamical systems.CrossRefGoogle Scholar
[10]Shields, P.. The Theory of Bernoulli Shifts (Chicago Lectures in Mathematics). The University of Chicago Press, Chicago–London, 1973.Google Scholar
[11]Walters, P.. An Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79). Springer, New York, 1982.CrossRefGoogle Scholar
[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. 189–196.CrossRefGoogle Scholar