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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)
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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.

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

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