BGU Probability and Ergodic Theory (PET) seminar Ical Atom

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.

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.

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.