We will discuss the non-stationary version of the Central Limit Theorem for random products of SL(2, R) matrices. The stationary versions were obtained previously by Tutubalin, Le Page, and Benoist-Quint. In all the previous works the assumption that the random matrices are identically distributed was used in a crucial way. We will explain how the recent results on the rate of growth of non-stationary products of random matrices can be used to overcome this restriction. The talk is based on a project joint with A. Gorodetski and G. Monakov. 

We consider the problem of classifying Kolmogorov automorphisms (or K-automorphisms for brevity) up to isomorphism or up to Kakutani equivalence. Within the collection of measure-preserving transformations, Bernoulli shifts have the ultimate mixing property, and K-automorphisms have the next-strongest mixing properties of any widely considered family of transformations. Therefore one might hope to extend Ornstein’s classification of Bernoulli shifts up to isomorphism by a numerical Borel invariant to a classification of K-automorphisms by some type of Borel invariant. We show that this is impossible, by proving that the isomorphism equivalence relation restricted to K-automorphisms, considered as a subset of the Cartesian product of the set of K-automorphisms with itself, is a complete analytic set, and hence not Borel. Moreover, we prove this remains true if we restrict consideration to K-automorphisms that are also C∞ diffeomorphisms. In addition, all of our results still hold if “isomorphism” is replaced by “Kakutani equivalence”. This shows in a concrete way that the problem of classifying K-automorphisms up to isomorphism or up to Kakutani equivalence is intractable. These results are joint work with Philipp Kunde.

A variety of questions and results on Cantor sets revolved around the Minkowski sums of Cantor sets and the topological structure or Hausdorff dimension of these sumsets.  For example, Shmeling and Schmerkin showed that given an increasing sequence {x_n} bounded by 0 and 1, there exists a Cantor set C such that x_n is the Hausdorff dimension of C added to itself n times. 

Given any integer n, we will provide a construction for a Cantor set with zero logarithmic capacity such that the Cantor set added to itself n times is a single interval, while a sum of any smaller number of copies of that set is still a Cantor set.

For this last talk in the series, I will discuss the details of how spectrum of the Schrödinger operator can be defined, and the proof that the spectrum can consist of an infinite number of intervals.