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

## 2022–23–B meetings

### Upcoming Meetings

Date
Title
Speaker
Abstract
Mar 30 TBA Thomas Ng (Technion - Israel Institute of Technology)
Apr 6 Passover break Passover break
Apr 13 TBA Nir Lazarovich (Technion - Israel Institute of Technology)
Apr 20 Mixing sequences for non-mixing locally compact Abelian groups actions El Houcein El Abdalaoui (CNRS-Université de Rouen Normandie)

Mixing is an important spectral property of dynamical systems and it can be described concretely. But, “In general a measure preserving transformation is” only “mixing” along a sequence of density one, by the Rhoklin-Halmos theorem. On the other hand, mixing on some sequences implies mixing. Formally, the mixing can be defined by demanding that the ergodic averages along any increasing sequence converge in mean, thanks to the Blum-Hanson theorem. In my talk, I will present my recent joint contribution with Terry Adams to Bergelson’s question asked online during the Lille conference 2021: Does mixing on the squares imply mixing? We first obtain a characterization of a sequence for which mixing on it implies mixing. We further establish that there are non-mixing maps that are mixing on appropriate sequences. We extend also our results to the group action with the help of the Host-Parreau characterization of the set of continuity from Harmonic Analysis. We further extended our result to the Real line action. As a open question, we ask pour extension our our result to the case of non-commutative case and specially Heisenberg group action.

Apr 27 TBA Zemer Kosloff (The Hebrew University of Jerusalem)
May 4 TBA Daren Wei (The Hebrew University of Jerusalem)
May 11 TBA
May 18 TBA Jon Aaronson (Tel Aviv University)
May 25 Shavuot - holiday Shavuot - holiday
Jun 1 TBA
Jun 8 TBA
Jun 15 TBA
Jun 22 TBA Amir Algom (University of Haifa)

### Past Meetings

Date
Title
Speaker
Abstract
Mar 16 A representation of Out(Fn) by counting subwords of cyclic words Noam Kolodner (Tel Aviv University)

We generalize the combinatorial approaches of Rapaport and Higgins–Lyndon to the Whitehead algorithm. We show that for every automorphism φ of a free group F and every word u∈F there exists a finite multiset of words Su,φ satisfying the following property: For every cyclic word w, the number of times u appears as a subword of φ(w) depends only on the appearances of words in Su,φ as subwords of w. We use this fact to construct a faithful representation of Out(Fn) on an inverse limit of Z-modules, so that each automorphism is represented by sequence of finite rectangular matrices, which can be seen as successively better approximations of the automorphism.

Mar 16, 14:00–15:00 Amenability is equivalent to the invariant random order extension property on groups Andrei Alpeev (The Weizmann Institute of Science)

Classical Szpilrajn theorem states that any partial order could be extended to a linear order. An invariant random order (IRO) on a countable group is an invariant under the shift-action probability measure on the space of all partial orders on the group. It is natural to ask whether the invariant analog of Szpilrajn theorem, the invariant random order extension property, holds for IRO’s. This property is easy to demonstrate for amenable groups. Recently, Glasner, Lin a Meyerovitch gave a first example where this property fails. Based on their construction, I will show that the IRO extension property fails for all non-amenable groups.

Mar 23 Postponed for later in the semester: Approximation of Diagonally Invariant measure by Tori Measures. Yuval Yifrach (Technion - Israel Institute of Technology)

We consider the family of periodic measures for the full diagonal action on the space of unimodular lattices. This family is important and natural due to its tight relation to class groups in number fields. We show that many natural families of measures on the space of lattices can be approximated using this family (in the weak sense). E.g., we show that for any 0<c\leq 1, the measure cm_{X_n} can be approximated this way, where m_{X_n} denotes the Haar probability measure on X_n. Moreover, we show that non ergodic measures can be approximated. Our proof is based on the equidistribution of Hecke neighbors and on constructions of special number fields. We will discuss the results, alternative ways to attack the problem, and our method of proof. This talk is based on a joint work with Omri Solan.