It is well-known that the entropy of an unknown stationary source can be
consistently estimated using an estimator based on the lengths of long
repeated sections of text. I will discuss a method for detecting change
points in data sources based on similar information theory-based
quantities. I will present the results of some simulations and
theoretical results based on properties of the typical set which justify
how successful this method can be

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.

Extended Harper's model arises in a tight binding description of 2
dimensional crystal layers subject to an external magnetic field. From a
dynamical point of view the model provides an example for a quasi-periodic
analytic Jacobi-cocyle with singularities.

In the first part of the talk, we show how to extend (and with what
limitations)
Avila's global theory of analytic SL(2,C) cocycles to families of cocycles
with singularities. In particular, we shall introduce a strategy of
computing the Lyapunov exponent valid for any analytic cocycle with
possible singularities.

As an application this allows to determine the Lyapunov exponent for
extended Harper's model, for all values of parameters, which so far did
not even exist on a heuristic level in physics literature.

In the second part of our talk, results on the spectral analysis of the
extended Harper's model will be presented.