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DCDS

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.

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