***Special Dynamical Systems and Ergodic Theory Seminar***

Furstenberg's proof of Szemeredi's theorem introduced the Correspondence Principle, a general technique for translating a combinatorial problem into a dynamical one. While the original formulation suffices for certain patterns, including arithmetic progressions and some infinite configurations, higher order generalizations have required refinements of these tools. We discuss the new techniques introduced in joint work with Moreira, Richter, and Robertson that are used to show the existence of infinite patterns in large sets of integers.

I will talk about some results on regularity of stationary measures of random dynamical systems. Under the assumption of absence of almost sure invariant measures we prove Holder continuity or log-Holder continuity of all stationary measures depending on estimates for the regularity of random maps. The methods I am going to present can also be used in the non-stationary case. This talk is based on a recent joint work with Anton Gorodetski and Victor Kleptsyn.

This is a "problem solving session" in a series that we plan to continue this year, aimed at graduate students who would like to get familiar with some aspects of dynamical systems. This set of problems covers basic examples of symbolic dynamical systems, including subshifts of finite type etc.. No preliminary background is expected from the participants. Everybody is welcome!

This is the first "problem solving session" in a series that we plan to continue this year, aimed at graduate students who would like to get familiar with some aspects of dynamical systems. The first set of problems covers basic properties of topological dynamical systems (minimality, recurrence, etc.). No preliminary background is expected from the participants. Everybody is welcome!

The classical Furstenberg Theorem states that the norm of a product of random i.i.d. matrices (under very mild assumptions) grows exponentially: almost surely one has
\lim_n (1/n) \log |A_n…A_1| = \lambda > 0.
My talk will be devoted to our recent work with Anton Gorodetski, where we have considered an analogous setting with A_j being independent, but no longer identically distributed. In such a setting it is natural to expect an (accordingly modified) version of the Furstenberg Theorem to hold. And indeed, we show that (again, under some mild assumptions on the distributions of A_j) there exists a deterministic sequence L_n such that

\liminf (1/n) L_n >0

 and almost surely

\lim (1/n) [\log |A_n…A_1| - L_n] = 0.

Moreover, there is an analogue of the Large Deviations Theorem.

The difficulty here is that in the nonstationary setting one cannot use the usual tools of the dynamical systems theory (stationary measure, ergodic theorem, etc.). I will discuss the proof of above results, as well as the general intuition of survival in the nonstationary world.