פעילויות השבוע

We show that the orbit equivalence relation of an action of a hyperbolic group on its Gromov boundary has finite Borel asymptotic dimension. As a corollary, that recovers the theorem of Marquis and Sabok which states that this orbit equivalence relation is hyperfinite.

A connection between the deformation rate of a small set boundary in the phase space of a dynamical system and the metric entropy of the system was claimed (not too rigorously) in physics literature.

Rigorous results were obtained by B. Gurevich for discrete time Markov shifts and later generalized for synchronized systems by me. Further, such a connection was established in joint work by B. Gurevich and S. Komech for Anosov diffeomorphisms, and for suspension flows in joint work by B. Gurevich, S. Komech and A. Tempelman. For symbolic dynamical systems, we estimate deformation rate in terms of an ergodic invariant measure, while for Anosov systems we use the volume. We will present specific details of our approach.

This talk is devoted to the study of some spectral properties of the Almost Mathieu Operator – a one-dimensional discrete Schrödinger operator with potential generated by an irrational rotation with angle \alpha (called the frequency). The spectral properties of the Almost Mathieu operator depend sensitively on the arithmetic properties of the frequency. If the frequency is poorly approximated by rationals, the spectral properties are as nice as one would expect.

The focus of this talk will be on the complementary case of well-approximated frequencies, in which the state of affairs is completely different. We show that in this case several spectral characteristics of the Almost Mathieu Operator can be as poor as at all possible in the class of all discrete Schrödinger operators. For example, the modulus of continuity of the integrated density of states (that is, of the averaged spectral measure) may be no better than logarithmic (for comparison, for poorly approximated frequencies the integrated density of states satisfies a Hölder condition). Other characteristics to be discussed are the Hausdorff measure of the spectrum and the non-homogeneity of the spectrum (as a set).

Based on joint work with A. Avila, Y. Last, and Q. Zhou