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.

Consider the critical percolation problem on the hexagonal lattice: each of (tiny) hexagons is independently declared «open» or «closed» with probability (1/2) — by a fair coin tossing. Assume that on the boundary of a simply connected domain four points A,B,C,D are marked. Then either there exists an «open» path, joining AB and CD, or there is a «closed» path, joining AD and BC (one can recall the famous «Hex» game here). Cardy’s formula, rigorously proved by S. Smirnov, gives an explicit value of the limit of such percolation probability, when the same smooth domain is put onto lattices with smaller and smaller mesh. Though, a next natural question is: what if more than four points are marked? And thus that there are more possible configurations of open/closed paths joining the arcs? 

In our joint work with M. Khristoforov we obtain the answer as an explicit integral for the case of six marked points on the boundary, passing through Fuchsian differential equations, Riemann surfaces, and Riemann-Hilbert problem. We also obtain a generalization of this answer to the case when one of the marked points is inside the domain (and not on the boundary).

We study Sturmian dynamical systems and the spectrum of Schrödinger operators associated with these systems. Plotting the spectrum for each dynamical system gives rise to the so-called Kohmoto butterfly. In this talk we will discuss this butterfly, compare it with the Hofstadter butterfly and draw connections to so-called Farey numbers.