Abstract:
The talk is based on a joint wofk with P.Merka, O.Paris-Romaskevich, A.Skripchenko and P.Hubert. The Novikov gsaket is a self-similar subset of an octahedron that parameterizes several families of dynamical systems related to each other. In particular, it can be used to describe a set of chaotic regimes in Novikov's problem on plane sections of 3-periodic surfaces in the case of a Fermi surface of genus three obeying central symmetry. (This is where the name comes from.) It also parameterizes some tiling billiards and interval exchange transformations with flips, distinguishing minimal ones. This set is fractal, i.e., it has Hausdorff dimension strictly less than three and, therefore, zero Lebesgue measure. However, it can be endowed with a probability measure in a natural way, with respect to which almost all corresponding exchange transformations with flips are strictly ergodic.
