Some Remarks on Novikov’s Problem for Foliations on Surfaces and Arnoux-Rauzy Interval Exchange Transformations

I will discuss some questions related to Novikov’s problem for foliations on surfaces.

The problem is the following. Let M be a 3-periodic surface and H a plane intersecting M. Which kind of curves are realized as the intersection of M and H?

This problem was formulated by Sergey Petrovitch Novikov in the 80’s. He conjectured that the « trivial » situations (periodic and integrable) are generic. I will discuss the simplest situation when M is very symmetric. In this case, with Dynnikov and Skripchenko, we prove that the first return of the foliation induced by H is an Arnoux-Rauzy interval exchange transformation. I will give some properties of these maps (results with Arnoux, Cassaigne and Ferenczi). In the most interesting situation, I will mention a work in progress with Dynnikov, Mercat, Paris-Romaskevich and Skripchenko which is supposed to solve Novikov’s conjecture.

The talk will be accessible to a broad audience.