BGU Probability and Ergodic Theory (PET) seminar Ical Atom

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.

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.

Combinatorial nonpositive curvature of CAT(0) cube complexes plays a surprising role both in topological characterization of hyperbolic 3-manifolds and also in studying algebraic properties of random groups.
With Einstein, Krishna MS, Montee, and Steenbock, we introduce a new model for random quotients of free products that generalizes Gromov’s destiny model. I will discuss challenges that arise in this new setting, connections to work of Futer-Wise and Martin-Steenbock on cubulating quotients, as well as applications to residual finiteness using recent work of Einstein and Groves on relative cubulation.

We show that isomorphic finite index subgroups of non-elementary hyperbolic groups must have the same index. In this talk I will present the tools and ideas of the proof.

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 will discuss a joint work with Dalibor Volny where we show that for every ergodic and aperiodic probability preserving transformation and α∈(0,2) there exists a function whose associated time series is in the standard domain of attraction of a non-degenerate symmetric α-stable distribution.

Two flows are said to be Kakutani equivalent if one is isomorphic to the other after time change, or equivalently if there are Poincare sections for the flows so that the respective induced maps are isomorphic to each other. Ratner showed that if $G=\operatorname{SL}(2,\mathbb{R})$ and $\Gamma$ is a lattice in $G$, and if $u_t$ is a one parameter unipotent subgroup in $G$ then the $u_t$ action on $G/\Gamma$ equipped with Haar measure is loosely Bernoulli, i.e.\ Kakutani equivalent to a circle rotation. Thus any two such systems $(\operatorname{SL}(2,\mathbb{R})/\Gamma_i, u_t, m_i)$ are Kakutani equivalent to each other. On the other hand, Ratner showed that if $G=\operatorname{SL}(2,\mathbb{R})\times \operatorname{SL}(2,\mathbb{R})$ and $\Gamma$ is a reducible lattice, and $u_t$ is the diagonally embedded one parameter unipotent subgroup in $G$, then $(G/\Gamma, u_t, m)$ is not loosely Bernoulli.

We show that in fact in this case and many other situations one cannot have Kakutani equivalence between such systems unless they are actually isomorphic.

This is a joint work with Elon Lindenstrauss.

I will discuss a random walk on a metric graph, that is, on a one-dimensional cell complex. The main difference from the often considered case is that the endpoint of a walk can be any point on an edge of a metric graph and not just one of the vertices. Let a point start its motion along the path graph from a hanging vertex at the initial moment of time. The passage time for each individual edge is fixed. At each vertex, the point selects one of the edges for further movement with some nonzero probability. Backward turns on the edges are prohibited in this model. One could find asymptotics for the number N(T) of possible endpoints of such a random walk as the time T increases, i.e. number of all possible lengths of paths on metric graph that not exceed T. Solutions to this problem, depending on the type of graph, are associated with different problems of number theory. An overview of the results, which depend on the arithmetic properties of lengths, will be given as well as review of open problems.

We show that the orbit equivalence relation of an action of a hyperbolic group on its Gromov boundary has finite Borel asymptotic dimension. As a corollary, that recovers the theorem of Marquis and Sabok which states that this orbit equivalence relation is hyperfinite.

A connection between the deformation rate of a small set boundary in the phase space of a dynamical system and the metric entropy of the system was claimed (not too rigorously) in physics literature.

Rigorous results were obtained by B. Gurevich for discrete time Markov shifts and later generalized for synchronized systems by me. Further, such a connection was established in joint work by B. Gurevich and S. Komech for Anosov diffeomorphisms, and for suspension flows in joint work by B. Gurevich, S. Komech and A. Tempelman. For symbolic dynamical systems, we estimate deformation rate in terms of an ergodic invariant measure, while for Anosov systems we use the volume. We will present specific details of our approach.

An analytic endomorphism of the unit disk is called an inner function if it’s boundary limit defines a transformation of the circle - which is necessarily Lebesgue nonsingular. I’ll review the ergodic theory of inner functions & then present results on:

– their structure;

– spectral gaps for their transfer operators; and

– a conditional central limit theorem;

all recently obtained with Mahendra Nadkarni.

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.

To any group, and more generally a C*-algebra, is associated its simplex of traces. The extreme points are called characters, which are a central notion in harmonic analysis. For a Kazhdan group, the simplex of traces is Bauer - the extreme points are closed. For the free group Fn the simplex of traces is Poulsen - the extreme points are dense. What about the simplex of Out(Fn)-invariant traces on Fn (n>3)? Is it Bauer, Poulsen or something in between? What about free products of finite groups, and free product of matrix algebras? Some answers and proofs will be provided, after an introduction on traces and characters. The talk is based on works with Levit, Orovitz and Slutsky.

Let $\Gamma=G_1*G_2*\dots *G_r$ be a free product of a finite number of finite groups and a finite number of copies of the infinite cyclic group. We sample uniformly at random an action of $\Gamma$ on $N$ elements. In this talk, we will discuss a few tools we developed to help answer some natural questions involving the configuration described above, such as: For $\gamma\in \Gamma$, what is the expected number of fixed points of $\gamma$ in the action we sampled? What is the the typical behavior of the cycle structure of the permutation corresponding to $\gamma$ etc.

This is a joint with Doron Puder.

Kaufman (1984) and later Mosquera-Shmerkin (2018) showed that Bernoulli convolutions exhibit fast Fourier decay when perturbed by a smooth non-linear map. This is remarkable, since by a classical Theorem of Erdos (1939) many Bernoulli convolutions don’t have Fourier decay at all. We will present an extension of this result to all self-similar measures: Any smooth non-linear perturbation of a self-similar measure enjoys fast (polynomial) Fourier decay. Joint with Yuanyang Chang, Meng Wu, and Yu-Liang Wu.

The Coupon Collector’s Problem (CCP) reads as follows: how many drawings are needed on average in order to complete a collection of $n$ types of coupons, if at each step a single coupon is drawn uniformly randomly with replacement, independently of all the other drawings? This problem was introduced by De-Moivre over 300 years ago. We will discuss about a generalization of the problem, where instead of drawing a single coupon each time we draw a ``package” of coupons of size $s$ and ask the following question: how does the distribution over the collection of possible .packages affect the expected number of drawings needed to complete a collection?

Despite their potentially significant and dramatic consequences, large deviations in chaotic dynamics have been studied very little, with few existing theoretical results. We study large deviations of series of finite lengths N generated by chaotic maps. The distributions generally display an exponential decay with N, associated with large-deviation (rate) functions. We calculate the exact rate functions analytically for the doubling, tent, and logistic maps, and numerically for the cat map. In the latter case, we uncover a remarkable singularity of the rate function that we interpret as a second order dynamical phase transition. Furthermore, we develop a numerical tool for efficiently simulating atypical realizations of sequences if the chaotic map is not invertible, and we apply it to the tent and logistic maps. Our research lays the groundwork for the study of unusual trends of long duration in chaotic systems, such as heatwaves or droughts in climate models, or unusual mean growth rate of a pandemic over a long period of time. The talk is based on the recent work [N. R. Smith, Phys. Rev. E 106, L042202 (2022)].