Echanges d'intervalles non topologiquement faiblement mélangeants

Abstract

In this paper, we prove a criterion for existence of continuous non constant eigenfunctions for interval exchange transformations, that is for non topologically weak mixing. We first construct, for any m>3, uniquely ergodic interval exchange transformations of rank 2 with irrational eigenvalues associated to continuous eigenfunctions, so that are not topologically weak mixing, this answers to a question of Ferenczi and Zamboni. Then, we construct, for any even number m>3, interval exchange transformations of rank 2 with both irrational eigenvalues (associated to continuous eigenfunctions) and non trivial rational eigenvalues (associated to piecewise continuous eigenfunctions). Moreover these examples can be chosen to be either uniquely ergodic or non minimal.