BGU Probability and Ergodic Theory (PET) seminar Ical Atom

Let \alpha be an irrational number and let J be a sub interval of [0,1]. The discrepancy sequence of J is D(N), where

Weyl’s Equidistribution Theorem says that D(N)=o(N). But this sequence is not necessarily bounded.

I will characterize the irrationals \alpha of bounded type, for which the discrepancy sequence of the interval [0,1/2] is equidistributed on (1/2)Z . This is joint work with Dima Dolgopyat.

Temporo-spatial differentiations are ergodic averages on a probabilistic dynamical system $(X, \mu, T)$ taking the form $\left( \frac{1}{\mu(C_k)} \int_{C_k} \frac{1}{k} \sum_{j = 0}^{k - 1} T^j f \mathrm{d} \mu \right)_{k = 1}^\infty $ where $C_k \subseteq X$ are measurable sets of positive measure, and $f \in L^\infty(X, \mu)$. These averages combine both the dynamics of the transformation and the structure of the underlying probability space $(X, \mu)$. We will discuss the motivations behind studying these averages, results concerning the limiting behavior of these averages and, time permitting, discuss generalizations to non-autonomous dynamical systems. Joint work with Idris Assani.

In studying higher dimensional Schrödinger operators of quasicrystals, one is lead to find suitable periodic approximations. This means in particular that the spectrum converges as a set to the limiting spectrum. It turns out that for this to hold, the convergence of the underlying dynamical systems is exactly what is needed. This is the starting point of the present talk.

We focus on aperiodic subshifts defined through symbolic substitutions. These substitution subshifts provide models of aperiodic ordered systems. We find natural sequence candidate of subshifts to approximate the aforementioned substitution subshift. We characterize when these sequences converge, and if so at what asymptotic rate. Some well-known examples of substitution subshifts are discussed during the talk. We will also discuss the motivation for this characterization, arising from an attempt to study higher dimensional quasi-crystals. This is based on a Joint work with Ram Band, Siegfried Beckus and Felix Pogorzelski.

In recent years, the study of measure preserving and stationary actions of Lie groups and hyperbolic groups have produced many geometric consequences. This talk will continue the tradition. We will show that stationary actions of hyperbolic groups have large critical exponent, namely exponential growth rate more than half of entropy divided the drift of the random walk.

This can be used to prove an interesting geometric result: if the bottom of the spectrum of the Laplacian on a hyperbolic n manifold M is equal to that of its universal cover (or equivalently the fundamental group has exponential growth rate at most (n-1)/2) then M has points with arbitrary large injectivity radius.

This is (in some sense the optimal) rank 1 analogue of a recent result of Fraczyk-Gelander which asserts that any infinite volume higher rank locally symmetric space has points with arbitrary large injectivity radius.

This is joint work with Arie Levit.

Given a non elementary locally compact hyperbolic group G equipped with a left invariant metric d one can define a measure on the Gromov boundary called the Patterson Sullivan measure associated to d. This measure is non singular with respect to the G action and contains geometric information on the metric. I will discuss the koopman representations of these actions and sketch a proof of their irreducibility and classification (up to unitary equivalence), generalizing works of Garncarek in the discrete case. I will also describe connections with a recent work of Caprace, Kalantar and Monod on the type I property for hyperbolic groups.

Ratio-limit boundaries were first studied for their applications to Toeplitz C-algebras of random walk, but are also interesting in their own right for measuring new types of behavior at infinity. For the purpose of describing Toeplitz C-algebras of random walks, new boundaries need to be identified in more precise terms. One such boundary is the so-called space-time Martin boundary, as studied by Lalley for random walks on the free group.

In this talk we will discuss ratio-limit boundaries and some work in progress on space-time Martin boundaries of random walks on discrete groups. The space-time Martin boundary is related to the notion of stability studied by Picardello and Woess, which elucidates potential descriptions of the space-time Martin boundaries for random walks on \mathbb{Z}^d and on hyperbolic groups.