# American Institute of Mathematical Sciences

2010, 27(3): 1079-1091. doi: 10.3934/dcds.2010.27.1079

## Non topologically weakly mixing interval exchanges

 1 Department of Mathematics, Faculty of Sciences of Bizerta, 7021 Jarzouna Tunisia, Springfield, MO 65801-2604, United States

Received  January 2009 Revised  February 2010 Published  March 2010

In this paper, we prove a criterion for the existence of continuous non constant eigenfunctions for interval exchange transformations which are non topologically weakly mixing. We first construct, for any $m>3$, uniquely ergodic interval exchange transformations of Q-rank $2$ with irrational eigenvalues associated to continuous eigenfunctions which are not topologically weakly mixing; this answers a question of Ferenczi and Zamboni [5]. Moreover we construct, for any even integer $m \geq 4$, interval exchange transformations of Q-rank $2$ with both irrational eigenvalues (associated to continuous eigenfunctions) and non trivial rational eigenvalues (associated to piecewise continuous eigenfunctions); these examples can be chosen to be either uniquely ergodic or non minimal.
Citation: Hadda Hmili. Non topologically weakly mixing interval exchanges. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1079-1091. doi: 10.3934/dcds.2010.27.1079
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