We will be looking at the trace map of the discrete Schrodinger operator with potential given by the period doubling sequence. It is known that for any positive coupling constant the spectrum of the corresponding operator is a Cantor set of Lebesgue measure zero. We are interested in the structure of the spectrum for small coupling constant, specifically the Hausdorff dimension and thickness.

The structure of Anosov-Katok example (in fact, this is a series of examples that can be constructed using similar techniques) will be presented. This is a way to build a smooth realization for several classes of measure preserving transformations.

The general belief is that attractors of diffeomorphisms of smooth manifolds either have measure zero, or coincide with the phase space. We prove that in the space of diffeomorphisms of a manifold with boundary onto itself there exists an open set (with at most a countable number of hypersurfaces deleted) such that any map from this set has a thick attractor: an attractor that has positive Lebesgue measure together with its complement. The result heavily relies upon the following two: ergodic theorems about the Hausdorff dimension of "exclusive" sets of some particular hyperbolic maps (P.Saltykov; Yu.Ilyashenko); overcoming of the "Fubini nightmare" for some perturbations of partially hyperbolic diffeomorphisms (joint work with A.Negut). Methods developed go back to investigations of A.Gorodetski and the speaker started at late 90's.

In this talk we we will first examine the dynamical properties of the simplest form of a piecewise isometry in one dimension, the interval exchange tranformation. We will then generalize this concept to interval translation mappings, and examine their dynamical properties, and consider an example of a rank 3 ITM which is of infinite type.