Activities This Week

A matrix cocycle is a non-commutative counterpart of random walk. The counterpart of the ergodic theorem, describing the almost sure asymptotic behaviour to leading order, is given by the theory of random matrix products originating in the works of Furstenberg—Kesten, Furstenberg, and Oseledec. On the other hand, the spectral theory of random one-dimensional second-order operators leads to the study of cocycles depending on an additional real number (the spectral parameter), and, a priori, the theory is applicable for almost all (rather than all) values of the parameter. The focus of the talk will be on the exceptional sets, where different asymptotic behaviour occurs: particularly, we shall discuss their rôle in spectral theory and their topologic and metric properties, including a result resembling the Jarnik theorem on Diophantine approximation. Based on joint work with Ilya Goldsheid.

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.