Consider a sequence of independent and identically distributed SL(2, R) matrices. There are several classical results by Le Page, Tutubalin, Benoist, Quint, and others that establish various forms of the central limit theorem for the products of such matrices. In our work, we generalize these results to the non-stationary case. Specifically, we prove that the properly shifted and normalized logarithm of the norm of a product of independent (but not necessarily identically distributed) SL(2, R) matrices converges to the standard normal distribution under natural assumptions. A key component of our proof is the regularity of the distribution of the unstable vector associated with these products.
In this talk, we will study random dynamical systems of smooth surface diffeomorphisms. Aaron Brown and Federico Rodriguez Hertz showed that, in this setting, hyperbolic stationary measures have the SRB property, except when certain obstructions occur. Here, the SRB property essentially means that the measure is absolutely continuous along certain “nice” curves (unstable manifolds). In this talk, we want to understand conditions that guarantee that SRB stationary measures are absolutely continuous with respect to the Lebesgue measure of the ambient space. Our approach is inspired by Tsujii's “transversality” method, which he used to show Palis conjecture for partially hyperbolic endomorphisms. This is a joint work with Aaron Brown, Homin Lee and Yuping Ruan.
The geometry of non-arithmetic hyperbolic manifolds is mysterious in spite of how plentiful they are. McMullen and Reid independently conjectured that such manifolds have only finitely many totally geodesic hyperplanes and their conjecture was recently settled by Bader-Fisher-Miller-Stover in dimensions larger than 3. Their works rely on superrigidity theorems and are not constructive.
In this talk, we strengthen their result by proving a quantitative finiteness theorem for non-arithmetic hyperbolic manifolds that arise from a gluing construction of Gromov and Piatetski-Shapiro. Perhaps surprisingly, the proof relies on an effective density theorem for certain periodic orbits. The effective density theorem uses a number of ideas including Margulis functions, a restricted projection theorem, and an effective equidistribution result for measures that are nearly full dimensional. This is joint work with K. W. Ohm.
We will consider a nonstationary sequence of independent random elements of a compact metrizable group. Assuming a natural nondegeneracy condition we will establish a weak-* convergence to the Haar measure, Ergodic Theorem, and Large Deviation Type Estimate. In particular, we will prove a nonstationary analog of classical Ito-Kawada theorem and give a new alternative proof for the stationary case. We will also show that all the results can be carried over to the case of a composition of random isometries.
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.
Using recent results on dynamics of non-stationary random matrix products, we establish spectral and dynamical localization for 1D Schrodinger operators with potentials given by independent but not identically distributed random variables.
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.
I will give an overview of the most important results about stationary random walks on SL(k, R). We will talk about Lyapunov exponent and their properties, such as positivity of the top exponent, simplicity and regularity of the spectrum and others. We will also mention other limit theorems, such as central limit theorem and law of iterated logarithm. The talk is based on the monograph ``Random walks on groups and random transformations'' by Alex Furman.
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.