Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1111.2420

Measure-theoretic chaos (Chaos au sens de la mesure)

Tomasz Downarowicz·Yves Lacroix
View original source ↗Read source document ↗Export metadata ↓

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 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 Résumé} Nous introduisons de nouveaux invariants pour les systèmes dynamiques définis sur des espaces probabilisés standards, appelés respectivement {\rm chaos mesuré} et {\rm chaos$^+$ mesuré}. Ces notions sont des analogues du chaos topologique {\rm DC2} et de l'une de ses variantes, renforcée, que nous appelons {\rm DC}{\small$1 \tfrac12$}. Nous montrons d'une part que si un système dynamique topologique est chaotique (resp. chaotique$^+$) au sens da la mesure relativement à\ l'une de ses mesures invariantes ergodiques, alors il l'est du point de vue topologique au sens correspondant. Nous montrons que tout système ergodique d'entropie métrique positive est chaotique$^+$ au sens de la mesure (même en un sens plus fort, i.e. {\rm uniformément}). Nous donnons enfin un exemple de système dynamique topologique d'entropie nulle qui présente pour l'une de ses mesures invariantes ergodiques un chaos$^+$ mesuré uniforme.

Explore related subjects

math.DSmath.PR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Tomasz Downarowicz, Yves Lacroix. 2012-06-20. Measure-theoretic chaos (Chaos au sens de la mesure). https://doi.org/10.1017/etds.2012.117

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Admissible Fourier Lengths, KAM Reducibility, and Spectral Applications

We develop a perturbative KAM reducibility theory for one-frequency $\mathrm{SL}(2,\mathbb{R})$ cocycles based on an admissible Fourier length $\ell$. The regularity relevant to the iteration is measured by positive adapted Fourier width rather than ordinary smoothness in the Euclidean length $|n|$. The same length governs Fourier decay, truncation and resonance scales, and the arithmetic condition controlling the small divisors. This framework contains the classical analytic and Gevrey settings, while non-monotone choices of $\ell$ allow classical nowhere differentiable Weierstrass-type perturbations and continuous perturbations outside every positive H\"older class. As spectral applications, we obtain purely absolutely continuous spectrum for every phase and $1/2$-H\"older continuity of the integrated density of states for the associated quasiperiodic Schr\"odinger operators. The Aubry dual has pure point spectrum for Lebesgue almost every dual phase, with eigenfunctions exponentially localized in the metric induced by $\ell$. We also construct nowhere differentiable quasiperiodic potentials with purely absolutely continuous Cantor spectrum.

math.DS↗
arXiv2026

Dynamics inside the attracting basins of some skew products

Polynomial skew products in $\mathbb{C}^2$ are maps of the form $F(z,w)=(P(z),Q(z,w))$, where $P$ and $Q$ are polynomials. Their local dynamics have been widely investigated. In this paper, we study the global dynamics inside Fatou components of some skew products. We consider all the inverse images in a Fatou component of a given point and use the Kobayashi metric to measure the distance between points. In the cases we consider, there are always arbitrarily large Kobayashi balls in the complement of these inverse sets.

math.DS↗
arXiv2026

Ergodicity of dynamical systems without uniqueness of orbits

Recently, there has been considerable interest in the study of non-deterministic dynamical systems. To analyze the chaotic behavior of such systems from a measure-theoretic viewpoint, it is desirable to consider ergodicity. However, the classical definition of ergodicity involves invariant sets, whose definition is not unique for non-deterministic dynamical systems. Thus, we are led to the question of which invariance yields an interesting definition of ergodicity. Here, we propose a definition based on the strong backward invariance and show that analogs of classical results hold. We also consider implications of the Birkhoff ergodic theorem for systems without uniqueness of orbits.

math.DS↗