Bounded complexity, mean equicontinuity and discrete spectrum - Laboratoire de Probabilités, Statistique et Modélisation Accéder directement au contenu
Article Dans Une Revue Ergodic Theory and Dynamical Systems Année : 2021

Bounded complexity, mean equicontinuity and discrete spectrum

Wen Huang
  • Fonction : Auteur
Jian Li
Leiye Xu
  • Fonction : Auteur
Xiangdong Ye
  • Fonction : Auteur

Résumé

We study dynamical systems that have bounded complexity with respect to three kinds metrics: the Bowen metric $d_{n}$ , the max-mean metric $\hat{d}_{n}$ and the mean metric $\bar{d}_{n}$ , both in topological dynamics and ergodic theory. It is shown that a topological dynamical system $(X,T)$ has bounded complexity with respect to $d_{n}$ (respectively $\hat{d}_{n}$ ) if and only if it is equicontinuous (respectively equicontinuous in the mean). However, we construct minimal systems that have bounded complexity with respect to $\bar{d}_{n}$ but that are not equicontinuous in the mean. It turns out that an invariant measure $\unicode[STIX]{x1D707}$ on $(X,T)$ has bounded complexity with respect to $d_{n}$ if and only if $(X,T)$ is $\unicode[STIX]{x1D707}$ -equicontinuous. Meanwhile, it is shown that $\unicode[STIX]{x1D707}$ has bounded complexity with respect to $\hat{d}_{n}$ if and only if $\unicode[STIX]{x1D707}$ has bounded complexity with respect to $\bar{d}_{n}$ , if and only if $(X,T)$ is $\unicode[STIX]{x1D707}$ -mean equicontinuous and if and only if it has discrete spectrum.
Fichier principal
Vignette du fichier
1806.02980v3.pdf (382.26 Ko) Télécharger le fichier
Origine Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-03939802 , version 1 (10-07-2024)

Identifiants

Citer

Wen Huang, Jian Li, Jean-Paul Thouvenot, Leiye Xu, Xiangdong Ye. Bounded complexity, mean equicontinuity and discrete spectrum. Ergodic Theory and Dynamical Systems, 2021, 41 (2), pp.494-533. ⟨10.1017/etds.2019.66⟩. ⟨hal-03939802⟩
9 Consultations
0 Téléchargements

Altmetric

Partager

Gmail Mastodon Facebook X LinkedIn More