SPECTRAL GAP PROPERTY AND STRONG ERGODICITY FOR GROUPS OF AFFINE TRANSFORMATIONS OF SOLENOIDS - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Ergodic Theory and Dynamical Systems Année : 2020

SPECTRAL GAP PROPERTY AND STRONG ERGODICITY FOR GROUPS OF AFFINE TRANSFORMATIONS OF SOLENOIDS

Résumé

Let X be a solenoid, that is, a compact finite dimensional connected abelian group with normalized Haar measure µ and let Γ → Aff(X) be an action of a countable discrete group Γ by continuous affine transformations of X. We show that the probability measure preserving action of Γ on (X,µ) does not have the spectral gap property if and only if there exists a p(Γ)-invariant proper subsolenoid Y of X such that the image of Γ in Aff(X/Y) is a virtually solvable group, where p : Aff(X) → Aut(X) is the canonical projection. When Γ is finitely generated or when X is the p-adic solenoid, the subsolenoid Y can be chosen so that the image Γ in Aff(X/Y) is virtually abelian. In particular, an action of a group by affine transformations on a solenoid has the spectral gap property if and only if this action is strongly ergodic.
Fichier principal
Vignette du fichier
Bekka-Francini-SpectralGap-Solenoids.pdf (323.61 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)
Loading...

Dates et versions

hal-01716937 , version 1 (25-02-2018)

Identifiants

Citer

Bachir Bekka, Camille Francini. SPECTRAL GAP PROPERTY AND STRONG ERGODICITY FOR GROUPS OF AFFINE TRANSFORMATIONS OF SOLENOIDS. Ergodic Theory and Dynamical Systems, 2020, 40 (5), pp.1180-1193. ⟨10.1017/etds.2018.114⟩. ⟨hal-01716937⟩
238 Consultations
98 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More