Extensions with shrinking fibers - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Ergodic Theory and Dynamical Systems Année : 2021

Extensions with shrinking fibers

Résumé

We consider dynamical systems $T: X \to X$ that are extensions of a factor $S: Y \to Y$ through a projection $\pi: X \to Y$ with shrinking fibers, i.e. such that $T$ is uniformly continuous along fibers $\pi^{-1}(y)$ and the diameter of iterate images of fibers $T^n(\pi^{-1}(y))$ uniformly go to zero as $n \to \infty$. We prove that every $S$-invariant measure has a unique $T$-invariant lift, and prove that many properties of the original measure lift: ergodicity, weak and strong mixing, decay of correlations and statistical properties (possibly with weakening in the rates). The basic tool is a variation of the Wasserstein distance, obtained by constraining the optimal transportation paradigm to displacements along the fibers. We extend to a general setting classical arguments, enabling to translate potentials and observables back and forth between $X$ and $Y$.
Fichier principal
Vignette du fichier
extensions.pdf (1.2 Mo) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-01961053 , version 1 (19-12-2018)
hal-01961053 , version 2 (07-02-2020)
hal-01961053 , version 3 (04-04-2020)

Identifiants

Citer

Benoit R. Kloeckner. Extensions with shrinking fibers. Ergodic Theory and Dynamical Systems, 2021, 41 (6), pp.1795 - 1834. ⟨10.1017/etds.2020.22⟩. ⟨hal-01961053v3⟩
90 Consultations
171 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More