Mathematics – Dynamical Systems
Scientific paper
2011-11-10
Mathematics
Dynamical Systems
24 pages
Scientific paper
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 chaoses {\rm DC2} and its slightly stronger version (which we denote by {\rm DC}{\small$1\tfrac12$}). We prove that: 1. If a \tl\ system is measure-theoretically (measure-theoretically$^+$) chaotic with respect to at least one of its ergodic measures then it is \tl ly {\rm DC2} ({\rm DC}{\small$1 \tfrac12$}) chaotic. 2. Every ergodic system with positive Kolmogorov--Sinai entropy is measure-theoretically$^+$ chaotic (even in a bit stronger uniform sense). We provide an example showing that the latter statement cannot be reversed, a system of entropy zero with uniform measure-theoretic$^+$ chaos. \bigskip \centerline{\bf R\'esum\'e} Nous introduisons de nouveaux invariants pour les syst\`emes dynamiques d\'efinis sur des espaces probabilis\'es standards, appel\'es respectivement {\rm chaos mesur\'e} et {\rm chaos$^+$ mesur\'e}. Ces notions sont des analogues du chaos topologique {\rm DC2} et de l'une de ses variantes, renforc\'ee, que nous appelons {\rm DC}{\small$1 \tfrac12$}. Nous montrons d'une part que si un syst\`eme dynamique topologique est chaotique (resp. chaotique$^+$) au sens da la mesure relativement \`a\ l'une de ses mesures invariantes ergodiques, alors il l'est du point de vue topologique au sens correspondant. Nous montrons que tout syst\`eme ergodique d'entropie m\'etrique positive est chaotique$^+$ au sens de la mesure (m\^eme en un sens plus fort, i.e. {\rm uniform\'ement}). Nous donnons enfin un exemple de syst\`eme dynamique topologique d'entropie nulle qui pr\'esente pour l'une de ses mesures invariantes ergodiques un chaos$^+$ mesur\'e uniforme.
Downarowicz Tomasz
Lacroix Yves
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