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.
Domaines
Systèmes dynamiques [math.DS]
Origine : Fichiers produits par l'(les) auteur(s)
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