Activities This Week
The degree of nonminimality is at most two (Special lecture) Online
May 8, 14:10—15:00, 2023, Department of mathematics, BGU, room -101
This is the third lecture from the Mini-Course Model theory of algebraic vector fields by Rahim Moosa. The first lecture is given as a Colloquium talk, and the details for the second one are here.
Abstract
In this final lecture I will sketch the proof that the degree of nonminimality of a finite rank type in DCF is at most two, and deduce as a consequence one of the theorems stated in Lecture 1.
Colloquium
Sets of non-Lyapunov behaviour for matrix cocycles
May 9, 14:30—15:30, 2023, Math -101
Speaker
Sasha Sodin (Queen Mary University of London)
Abstract
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.
BGU Probability and Ergodic Theory (PET) seminar
Borel asymptotic dimension for boundary actions of hyperbolic groups
May 11, 11:10—12:00, 2023, -101
Speaker
Petr Naryshkin (WWU Münster)
Abstract
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.
BGU Probability and Ergodic Theory (PET) seminar
Geometric approach to the Kolmogorov entropy
May 11, 14:00—15:00, 2023, -101
Speaker
Sergey Komech (The Institute for Information Transmission Problems)
Abstract
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.