In this thesis we investigate the ergodic properties of area-preserving flows which are constructed as suspension flows over interval exchange transformations under functions with logarithmic singularities. Interval exchange transformations (IETs) are piecewise isometries of an interval and suspension flows over IETs are a useful tool to describe different time parameterization of flows on surfaces. Logarithmic singularities arise considering locally Hamiltonian flows, in presence of generic saddles. We prove that, in the case of an asymmetric logarithmic singularity, such suspension flows are mixing or asymptotically uniformly distributed in space, for a full measure set of IETs. In particular, we formulate a "diophantine-like" sufficient condition on the IET. Rauzy, Veech and Zorich developed a renormalization algorithm for IETs which is a multidimensional generalization of the continued fraction algorithm for rotations. We use it to express the condition and prove that it has full measure from the integrability properties of the associated cocycle. In the case of a symmetric logarithmic singularity, we show that, for a full measure set of IETs, the suspension flows are weakly mixing , or have continuous spectrum. On the other hand, we can construct a class of examples of IETs for which the suspension flows are not mixing. One of the key ingredients in both cases, which is of separate interest, are estimates on the growth of the Birkhoff sums of functions with singularities of type 1/ x over a typical IET. These estimates provide quantitative grounds for the stretching of short transversal segments in the flow direction, which is at the origin of the geometric mechanism for mixing.
|Translated title of the contribution||Ergodic properties of some area-preserving flows|
|Number of pages||125|
|Publication status||Published - 2007|