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Measure-theoretic chaos

Published online by Cambridge University Press:  31 August 2012

TOMASZ DOWNAROWICZ  and
YVES LACROIX
TOMASZ DOWNAROWICZ
Affiliation:
Institute of Mathematics and Computer Science, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland (email: downar@pwr.wroc.pl)
YVES LACROIX
Affiliation:
Institut des Sciences de l’Ingénieur de Toulon et du Var, Avenue G. Pompidou, B.P. 56, 83162 La Valette du Var Cedex, France (email: yves.lacroix@univ-tln.fr)
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Abstract

We define new isomorphism invariants for ergodic measure-preserving systems on standard probability spaces, called measure-theoretic chaos and measure-theoretic$^+$ chaos. These notions are analogs of the topological chaos DC2 and its slightly stronger version (which we denote by $\text {DC}1\frac 12$). We prove that: (1) if a topological system is measure-theoretically (measure-theoretically$^+$) chaotic with respect to at least one of its ergodic measures then it is topologically DC2 $(\text {DC}1\frac 12)$ chaotic; (2) every ergodic system with positive Kolmogorov–Sinai entropy is measure-theoretically$^+$ chaotic (even in a slightly stronger uniform sense). We provide an example showing that the latter statement cannot be reversed, that is, of a system of entropy zero with uniform measure-theoretic$^+$chaos.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 1 , February 2014 , pp. 110 - 131
Copyright
Copyright © 2012 Cambridge University Press 

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