Colloquium

Metric Ergodicity and its applications in Rigidity Theory

 An action is called Metrically Ergodic (ME) if the acted space admits no equivariant maps to metric spaces. This is a far reaching strengthening of usual Ergodicity. For probability measure preserving actions, ME is equivalent to Weak Mixing, but in general it is different and could be thought of as a generalization of Mautner phenomenon. It is a remarkable fact that every locally compact second countable group admits an action which is both Amenable and Metrically Ergodic. For example, the celebrated fixed point theorem of Ryll-Nardzewski could be deduced easily from this fact. In another direction, it implies Super-Rigidity for Lattices in Products. It is also a main player in the proof of the Simplicity of the Lyapunov Spectra in various situations. In my talk I will provide a gentle introduction to ME and discuss its applications.

The talk is based on a joint work with Alex Furman.

Some variants of the Rogers integral formula

The Rogers integral formula from the 1950's, which provides statistical information on points of the random lattice, have recently become a popular device in number theory and dynamics. With the revived interest in the formula also came its numerous variants and extensions. In this talk, I would like to discuss a few such formulas by myself, one concerning the rational points of the Grassmannians, and another generalizing the Rogers formula over the adeles of number fields. One immediate application is the non-Poissonian behavior of certain families of random sets of points, unlike in the much-studied case of the random lattice points in R^n.

Farey Sequences for Thin Groups

The study of thin groups has become a hot topic at the centre of a great deal of modern research. Recently the measure theory on infinite hyperbolic groups has opened these groups up to the techniques of homogeneous dynamics. In this talk I will give a brief and incomplete survey of this development and then explain a recent result about the local statistics of orbits by these groups and the connection to local statistics of some sequences of numbers. If there is time I will explain the connections between these theorems, Diophantine approximation on fractals, Ford packings, and continued fractions.

Discrete stationary random subgroups and application to discrete subgroups of Lie groups

The notion of invariant random subgroups (IRS) has proven extremely useful during the last decade, particularly to the study of asymptotic invariants of lattices. However, the scope of problems that one can investigate when restricting to invariant measures (on the space of subgroups)is limited. It was recently realised that the notion of stationary random subgroups (SRS), which is much more general, is still extremely powerful and opens up new paths to attacking problems that previously seemed to be out of our reach. 

  In this talk, using the notion of SRS, I will explain a proof of the following conjecture of Margulis: Let G be a higher rank simple Lie group and Λ ⊂ G a discrete subgroup. Then the orbifold Λ\G/K has finite volume if and only if it has bounded injectivity radius. This is a far-reaching generalisation of the celebrated Normal Subgroup Theorem of Margulis, and while it is new even for subgroups of lattices, it is completely general.    

This is a joint work with Mikolaj Fraczyk.

Invariant random subgroups and applications to lattices

In mathematics in general, it is fruitful to allow randomness. Indeed, it is often easier to deal with random rather than deterministic objects. It seems miraculous, however, when we are able to say more about deterministic objects by considering them as random ones.

The notion of invariant random subgroups (IRS) has proven extremely useful  to the study of asymptotic invariants of lattices. In this talk I will demonstrate several results (old and new) about lattices in semisimple Lie groups which are proved by considering them as IRS.

Estimating the Hausdorff dimension of Julia sets of holomorphic correspondences

We consider a special class of holomorphic correspondences which generalizes the quadratic family and use some tools of thermodynamic formalism to estimate the Hausdorff dimension of the Julia sets of this family of holomorphic correspondences.

Uniformization of quasiconformal trees

Quasisymmetric maps are generalizations of conformal maps and may be viewed as a global versions of quasiconformal maps. Originally, they were introduced in the context of geometric function theory, but appear now in geometric group theory and analysis on metric spaces among others. The quasisymmetric uniformization problem asks when a given metric space is quasisymmetric to some model space. Here we consider this uniformization problem for certain trees. In particular, we consider the "continuum self-similar tree'' (CSST) and give necessary and sufficient conditions for another tree to be quasisymmetrically equivalent to the CSST. One motivation to study the CSST is that it is almost surely homeomorphic to the continuum random tree introduced by Aldous. This is joint work with Mario Bonk.

Subshifts of finite type with a hole: properties and applications

Dynamical systems can be broadly classified into closed and open systems. In a (traditional) closed system, the orbit of a point lies in the state space for all time, whereas in an open system, the orbit of a point may eventually escape from the state space through a hole. The notion of open dynamical systems was introduced by Pianigiani and Yorke in 1979, motivated by the dynamics of a ball on a billiard table with pockets. In this talk, we consider an irreducible subshift of finite type and study the average rate at which the orbits escape into the hole. This problem turns out to be an interesting application of a combinatorial problem which is counting the number of words of fixed length not containing any of the words from a fixed collection as subwords. We also present some applications of our results in computing the Perron eigenvalues and eigenvectors of any irreducible (0-1) matrix and obtaining a combinatorial expression for the Parry measure of a subshift of finite type.

Volume vs. complexity of hyperbolic groups

In this talk we will discuss the relation between the volume of a quotient X/G of a (Gromov) hyperbolic graph X by a group G acting freely and cocompactly on X, and the "complexity" of the group G. We will then show how to use this relation to study finite-index subgroups of cubulated hyperbolic groups.

Brolin’s theorem for finitely generated polynomial semigroups

In this talk, we give a description of a natural invariant measure associated with a finitely generated polynomial semigroup (which we shall call the Dinh--Sibony measure) in terms of potential theory. The existence of this measure follows from a very general result of Dinh--Sibony applied to a holomorphic correspondence that one can naturally associate with a semigroup of the above type. We are interested in a precise description of this invariant measure. This requires the theory of logarithmic potentials in the presence of an external field, which, in our case, is explicitly determined by the choice of a set of generators. Our result generalizes the classical result by Brolin. Along the way, we establish the continuity of the logarithmic potential for the Dinh--Sibony measure, which might be of independent interest. If time permits, we shall also present some bounds on the capacity and diameter of the Julia sets of such semigroups, which uses the F-functional of Mhaskar and Saff.