The seminar meets on Thursdays, 11:10-12:00, in -101

This Week


Lior Tenenbaum (Technion)

Periodic approximation of substitution subshifts

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.


2023–24–B meetings

Upcoming Meetings

Date
Title
Speaker
Abstract
May 30 Periodic approximation of substitution subshifts Lior Tenenbaum (Technion)

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.

Jun 13 TBA Gill Goffer (UCSD)
Jun 27 TBA Ilya Gekhtman (Technion)

Past Meetings

Date
Title
Speaker
Abstract
May 9 Higher Kazhdan Property and Unitary Cohomology of Arithmetic Groups Uri Bader (BGU)
May 16 Equidistribution of Discrepancy Sequences (Joint with Dolgopyat) Omri Sarig (Weizmann Institute of Science)

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

D(N):=the number of visits of n\alpha mod 1 to J for 1<n<N minus N J .

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.

May 23 Random temporo-spatial differentiations Adian Young (BGU)

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.