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

This Week


Place: Online

Yiftach Dayan (Technion)

Random walks on tori and an application to normality of numbers in self-similar sets.

We show that under certain conditions, random walks on a d-dim torus by affine expanding maps have a unique stationary measure. We then use this result to show that given an IFS of contracting similarity maps of the real line with a uniform contraction ratio 1/D, where D is some integer > 1, under some suitable condition, almost every point in the attractor of the given IFS (w.r.t. a natural measure) is normal to base D. (Joint work with Arijit Ganguly and Barak Weiss.)


2020–21–B meetings

Upcoming Meetings

Date
Title
Speaker
Abstract
May 20, In Online Random walks on tori and an application to normality of numbers in self-similar sets.Online Yiftach Dayan (Technion)

We show that under certain conditions, random walks on a d-dim torus by affine expanding maps have a unique stationary measure. We then use this result to show that given an IFS of contracting similarity maps of the real line with a uniform contraction ratio 1/D, where D is some integer > 1, under some suitable condition, almost every point in the attractor of the given IFS (w.r.t. a natural measure) is normal to base D. (Joint work with Arijit Ganguly and Barak Weiss.)

May 27, 10:00–10:45, In Online TBAOnline Doron Puder (Tel-Aviv University)
Jun 3, In Online Slow entropy of higher rank abelian unipotent actionsOnline Daren Wei (The Hebrew University)

We study slow entropy invariants for abelian unipotent actions U on any finite volume homogeneous space $G/\Gamma$. For every such action we show that the topological complexity can be computed directly from the dimension of a special decomposition of Lie(G) induced by Lie(U). Moreover, we are able to show that the metric complexity of the action coincides with its topological complexity, which provides a classification of these actions in isomorphic class. As a corollary, we obtain that the complexity of any abelian horocyclic action is only related to the dimension of G. This generalizes our previous rank one results from to higher rank abelian actions. This is a joint work with Adam Kanigowski, Philipp Kunde and Kurt Vinhage.

Jun 10, In Online TBAOnline Henna Koivusalo (University of Bristol)
Jun 17, In Online TBAOnline Eitan Bachmat (Ben-Gurion University)

Past Meetings

Date
Title
Speaker
Abstract
Mar 4, In Online TBAOnline
Mar 11, 16:00–17:00, In Online Effective equidistribution of horospherical flows in infinite volumeOnline Nattalie Tamam (University of California, San Diego)

Horospherical flows in homogeneous spaces have been studied intensively over the last several decades and have many surprising applications in various fields. Many basic results are under the assumption that the volume of the space is finite, which is crucial as many basic ergodic theorems fail in the setting of an infinite measure space. In the talk we will discuss the infinite volume setting, and specifically, when can we expect horospherical orbits to equidistribute. Our goal will be to provide an effective equidistribution result, with polynomial rate, for horospherical orbits in the frame bundle of certain infinite volume hyperbolic manifolds. This is a joint work with Jacqueline Warren.

Mar 18, In Online A multiplicative ergodic theorem for von Neumann algebra valued cocyclesOnline Yuqing Frank Lin (Ben-Gurion University)

Oseledets’ multiplicative ergodic theorem (MET) is an important tool in smooth ergodic theory. It may be viewed as a generalization of Birkhoff’s pointwise ergodic theorem where numbers are replaced by matrices and arithmetic means are replaced by geometric means. Starting from Ruelle in 1982, many infinite-dimensional generalizations of the MET have been produced; however, these results assume quasi-compactness conditions and so do not deal with continuous spectrum. In a different direction Karlsson-Margulis obtained a geometric generalization of the MET, which we apply in our work to obtain an MET with operators in von Neumann algebras with semi-finite trace. We do not assume any compactness conditions on the operators. Joint work with Lewis Bowen and Ben Hayes.

Mar 25 Passover break
Apr 1 Passover break
Apr 8 Holocaust Memorial Day
Apr 15 Memorial day for Israel’s fallen
Apr 22, In Online Random walks on finite partite simplicial complexesOnline Zohar Reizis (Ben-Gurion University)

Random walks on graphs (and their spectral analysis) is an extensively explored topic with many applications in pure math and computer science. Recently, there has been much interest (by both the math and the CS communities) in the study of random walks on simplicial complexes as a high dimensional generalization on random walks on graphs. In this talk, we consider the spectrum of random walks on finite partite simplicial complexes and show how with a general decomposition theorem on Hilbert spaces we can improve previous works. All the definitions will be given. This is a joint work with Izhar Oppenheim.

Apr 29, In Online About Borel and almost Borel embeddings for Z^d actionsOnline Nishant Chandgotia (The Hebrew University)

Krieger’s generator theorem says that all free ergodic measure preserving actions (under natural entropy constraints) can be modelled by a full shift. Recently, in a sequence of two papers Mike Hochman proved that this theorem can be strengthened: He showed that all free homeomorphisms of a Polish space (under entropy constraints) can be Borel embedded into the full shift. In this talk we will discuss some results along this line from a recent paper with Tom Meyerovitch and ongoing work with Spencer Unger.

With Meyerovitch, we established a condition called flexibility under which a large class of systems are almost Borel universal, meaning that such systems can model any free Z^d action on a Polish space up to a universally null set. The condition of flexibility covered a large class of examples including those of domino tilings and the space of proper 3-colourings (among many non-symbolic examples) and answered questions by Robinson and Sahin. However extending the embedding to include the null set is a daunting task and there are many partial results towards this. Using tools developed by Gao, Jackson, Krohne and Seward, along with Spencer Unger we were able to get Borel embeddings of symbolic systems (as opposed to all Borel systems) under assumptions very similar to flexibility. This answers questions by Gao and Jackson and recovered some results announced by Gao, Jackson, Krohne and Seward.

May 6, In Online Infinite volume and infinite injectivity radiusOnline Tsachik Gelander (Weizmann Institute)

We prove the following conjecture of Margulis. Let M=Λ\G/K be a locally symmetric space where G is a simple Lie group of real rank at least 2. If M has infinite volume then it admits injected contractible balls of any radius. This generalizes the celebrated normal subgroup theorem of Margulis to the context of arbitrary discrete subgroups of G and has various other applications. We prove this result by studying random walks on the space of discrete subgroups of G and analysing the possible stationary limits.

This is a joint work with Mikolaj Fraczyk.

May 13, In Online Around the Danzer problem and the construction of dense forests.Online Faustin Adiceam (The University of Manchester)

The still open Danzer problem (1965) asks for the existence of a set with finite density intersecting any convex body of volume one. It has so far attracted a considerable number of ideas revolving around many different areas (ergodic theory, probability, dynamical systems, Diophantine approximation, harmonic analysis, the theory of quasicrystals…).

After surveying the state of the art in this problem, we will focus our attention on the construction of so-called dense forests. These are discrete point sets emerging from the weakening of the volume constraint in Danzer’s question. The emphasis will be put on the effectiveness of such construction.

Based on joint work with Yaar Solomon and Barak Weiss.