I will describe recent advances in the Pugh-Shub stable ergodicity theory for partially hyperbolic diffeomorphisms. In particular, I consider two "competing" methods to show that a given partially hyperbolic diffeomorphism is stably ergodic (i.e., it is ergodic along with any of its sufficiently small perturbations). One of them relates the problem to to the global estimates of the action of the system along its central direction while another one deals with a more delicate estimates using Lyapunov exponents in the central direction.
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.
In this talk, we will begin with simple questions of the Diophantine Approximation Theory, for instance, how closely can a given irrational number x be approximated by a rational number r with denominator no larger than a fixed number? This will lead us to talk about the set known as the Lagrange Spectrum whose structure closely resembles the structure of the sum of dynamically defined Cantor sets, which are defined by an iterate system of expanding differentiable functions on intervals. We will construct two Cantor sets whose arithmetic sum is a uniformly contracting self-similar set. A local result on a sufficient condition for a uniformly contracting self-similar set to be of Lebesgue measure zero will be proven.
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.
We will review the recent (and not so recent) results on dynamics of piecewise isometries (especially piecewise translations), both in one and in higher dimensional case. Some interesting results (by Suzuki, Goetz, Zhuravlev, Boshernitzan, Bruin, Troubetzkoy, Buzzi) are known, but most of natural questions are still open. The main goal of the talk is to expose these open questions to potential researchers.
Cornell University, Independent University of Moscow, Moscow State University
Time:
Tuesday, December 7, 2010 - 3:00pm
Location:
RH 440R
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.