Sasha Skripchenko

The notion of systems of isometries was introduced by G. Levitt, D. Gaboriau and F. Paulin in 1994 as a natural generalization of interval exchange transformations. Later the same object was rediscovered by I. Dynnikov in connection with 3-dimensional topology. In my talk I will discuss some general properties of orbits of systems of isometries and open problems related to them.
In the second part of the talk I will illustrate some notions and statements from the first part on the baby example - some family of system of isometries that are described by 2 parameters. In particular, such systems of thin type form a fractal that also appears in symbolic dynamics under the name Rauzy gasket and corresponds to letter frequencies of ternary episturmian words. We prove that the Hausdorff dimension of the Rauzy gasket is strictly less then 2. This part is based on a joint work with A. Avila and P. Hubert.