Olga Paris-Romaskevich

Novikov’s problem and tiling billiards

In the beginning of the 80s, Masur and Veech independently proved that a generic interval exchange transformation is uniquely ergodic. At about the same time, on different sides of the iron curtain, mathematicians got interested in the ergodic properties of measured foliations restricted to some parametric classes. Their motivations were different but the same fractal objet appeared in both, the so-called Rauzy gasket.

The motivation from the Soviet side came from the question asked by Sergei Novikov related to the conductivity theory of monocrystals. It concerns classification of minimal foliations defined by the restriction of a closed 1-form on a surface of genus g embedded into the 3-torus.

I will give an introduction to Novikov’s problem and announce our result — its partial solution in the most physically relevant case. I will also point out a relationship of Novikov’s problem with an elementary model given by a dynamical system, tiling billiards.

The talk is based on an ongoing collaboration with Ivan Dynnikov, Pascal Hubert, Paul Mercat and Alexandra Skripchenko.