Author:
BEKKA BACHIR,GUIVARC’H YVES
Abstract
Abstract
Let
$S=\{p_1, \ldots , p_r,\infty \}$
for prime integers
$p_1, \ldots , p_r.$
Let X be an S-adic compact nilmanifold, equipped with the unique translation-invariant probability measure
$\mu .$
We characterize the countable groups
$\Gamma $
of automorphisms of X for which the Koopman representation
$\kappa $
on
$L^2(X,\mu )$
has a spectral gap. More specifically, let Y be the maximal quotient solenoid of X (thus, Y is a finite-dimensional, connected, compact abelian group). We show that
$\kappa $
does not have a spectral gap if and only if there exists a
$\Gamma $
-invariant proper subsolenoid of Y on which
$\Gamma $
acts as a virtually abelian group,
Publisher
Cambridge University Press (CUP)
Subject
Applied Mathematics,General Mathematics