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Spectral gap property and strong ergodicity for groups of affine transformations of solenoids

Part of: Ergodic theory Noncompact transformation groups

Published online by Cambridge University Press:  11 October 2018

BACHIR BEKKA  and
CAMILLE FRANCINI
BACHIR BEKKA
Affiliation:
Université de Rennes, CNRS, IRMAR-UMR 6625 Campus Beaulieu, F-35042 Rennes Cedex, France email bachir.bekka@univ-rennes1.fr, camille.francini@ens-rennes.fr
CAMILLE FRANCINI
Affiliation:
Université de Rennes, CNRS, IRMAR-UMR 6625 Campus Beaulieu, F-35042 Rennes Cedex, France email bachir.bekka@univ-rennes1.fr, camille.francini@ens-rennes.fr
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Abstract

Let $X$ be a solenoid, i.e. a compact, finite-dimensional, connected abelian group with normalized Haar measure $\unicode[STIX]{x1D707}$, and let $\unicode[STIX]{x1D6E4}\rightarrow \operatorname{Aff}(X)$ be an action of a countable discrete group $\unicode[STIX]{x1D6E4}$ by continuous affine transformations of $X$. We show that the probability measure preserving action $\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$ does not have the spectral gap property if and only if there exists a $p_{\text{a}}(\unicode[STIX]{x1D6E4})$-invariant proper subsolenoid $Y$ of $X$ such that the image of $\unicode[STIX]{x1D6E4}$ in $\operatorname{Aff}(X/Y)$ is a virtually solvable group, where $p_{\text{a}}:\operatorname{Aff}(X)\rightarrow \operatorname{Aut}(X)$ is the canonical projection. When $\unicode[STIX]{x1D6E4}$ is finitely generated or when $X$ is the $a$-adic solenoid for an integer $a\geq 1$, the subsolenoid $Y$ can be chosen so that the image $\unicode[STIX]{x1D6E4}$ in $\operatorname{Aff}(X/Y)$ is a virtually abelian group. In particular, an action $\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$ by affine transformations on a solenoid $X$ has the spectral gap property if and only if $\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$ is strongly ergodic.

Keywords

group actionsarithmetic algebraic dynamicsspectral gap propertystrong ergodicitysolenoids

MSC classification

Primary: 37A30: Ergodic theorems, spectral theory, Markov operators
Secondary: 22F10: Measurable group actions
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 5 , May 2020 , pp. 1180 - 1193
Copyright
© Cambridge University Press, 2018

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