| Résume||At the beggining of the 80's, M. Keane, H.Masur and W.Veech started the study of typical properties of interval exchange transformations proving that almost every such transformation is minimal and even uniquely ergodic. About the same time, S.Novikov's school and French mathematicians independently discovered very intriguing phenomena for some very special classes of measured foliations on surfaces and respective IETs that do not fit in the framework of Keane's theorem. For instance, minimality is exceptional in these families. A precise version of this statement is a conjecture by Novikov. The French and Russian constructions are very different ones. Nevertheless, in the most simple situation (surfaces of genus three with two singularities) it was recently observed that both foliations share the same type of properties. For instance, the space of minimal parameters is the same, called the Rauzy gasket. In the last several years we made a serious progress in understanding of ergodic properties of the foliations associated with the Rauzy gasket; moreover, we developed a precise vocabulary between different languages that were used to describe this class of foliations.
In my talk I plan to explain briefly these results that were mainly obtained jointly with Ivan Dynnikov and Pascal Hubert.
If the time allows, we will also talk about our new approach to the more general question - Novikov's conjecture in genus three for symmetric surfaces. It is a work in progress with I. Dynnikov, P. Hubert, P. Mercat and O. Romaskevich - Paris.|