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.

We will review several open problems in dynamical systems (mostly related to hyperbolic and partially hyperbolic dynamics) that can be used as a starting point for independent graduate student's research.

The one dimensional quantum Ising model is used in quantum statistical physics to model interracting particles on a discrete lattice. While the classical model (in one and two dimensions) has long been solved (its origin dates back to 1930's), its quasiperiodic analog (dating back about 25 years) is still a source of interesting problems. We shall discuss our solution to one such problem: we'll rigorously prove that the energy spectrum of the one dimensional quantum quasiperiodic Ising model is a Cantor set, as has been long believed, and discuss some of its properties.
This is the first in a series of two seminars dedicated to this topic. In this seminar we'll present the problem and set up the main ideas.