Spectral Properties of Interval Exchange Transformations

Abstract

This thesis consists of two main parts dealing with spectral properties of typical interval exchange transformations (IET for short). We establish a uniform lower bound (depending only on the Rauzy class of the corresponding permutation) on the local dimension of the spectral measures of Lipschitz observables for the typical IET of non-rotation type and use it to show a polynomial rate of weak mixing. For rotation-type IETs, however, the lower bound turns out to be not uniform in the spectral parameter which is responsible for a slower (logarithmic) rate of weak mixing in this case. We complement this result by showing that in the latter case, the rate of weak mixing cannot be better than logarithmic. This is done by providing suitable upper bounds for the associated twisted cocycle along a certain line segment of parameters. In the second part, we establish an explicit lower bound on the exponential growth rate of the twisted cocycle for typical parameters.