Speaker: Drew Ash

Title:Topological Speedups
Given a dynamical system (X,T) one can define a speedup of (X,T) as another dynamical system S : X → X where S = T^p(·) for some p : X → Z+. In 1985 Arnoux, Ornstein, and Weiss showed that given a pair of measure theoretic dynamical systems, one is isomorphic to a speedup of the other under the very mild condition that both transformations are aperiodic. In this talk I will give the setting and relevant definitions for what we mean by a topological speedup; then we will discuss a characterization theorem for speedups of minimal Cantor systems. This theorem is both a topological ana- log of the Arounx-Ornstein-Weiss result and a sort of “one-sided” version of a theorem of Giordano-Putnam-Skau on topological orbit equivalence.

 James T. Campbell

Title:  Recurrence for IP systems with polynomial wildcards.

We are interested in providing a new joint extension of the IP Szemerédi theorem of Furstenberg and Katznelson (what we call Theorem FK, [FK85]) and the polynomial Szemerédi theorem of V. Bergelson and A. Leibman [BL96]. That is, we wish to prove an IP, polynomial, multiple recurrence theorem. As a first step, we prove the more accessible single recurrence version of the intended result. In the talk, we begin with a survey of these (and related) results, move to a more detailed analysis of the meaning of Theorem FK, and then present our single recurrence result. This is joint work with Randall McCutcheon, to appear in the Transactions of the A.M.S.

[BL96] V.Bergelson and A.Leibman, Polynomial extensions of van de rwaerden’s and Szemerédi’s theorems, Journal of the AMS 9 (1996), 725–753.
[FK85] H. Furstenberg and Y. Katznelson, An ergodic Szemerédi theorem for IP-systems and combinatorial theory, J. d’Analyse Math. 45 (1985), 117–168.

: Yves Cornulier

Sofic profile 

Informally, a group is sofic if it has enough "quasi-actions" on finite sets. This is a weakening of being residually finite, which means that the group has enough actions on finite sets to separate the points. We introduce a quantitative version of being sofic, called sofic profile. Indeed, quasi-actions mean action defined up to a negligible subset indeterminacy points, and the sofic profile measures how small this indeterminacy set is. I will explain this in detail, mention basic results and natural questions.

Speaker: Maryam Hosseini

On Orbit Equivalence of Cantor Minimal Systems and their Continuous Spectrum

This is a joint work with Thierry Giordano and  David Handelman, we investigate about existence of a continuous eigenvalue for a Cantor minimal system, $(X, T)$,  with regards to its dimension group, $K^0(X,T)$. In this context, the notion of irrational mixability for dimension groups is introduced and some (necessary and) sufficient conditions for this property will be given. The main property of these dimension groups is the absence of   irrational values in the set of continuous spectrum of their realizations by Cantor minimal systems. Any realization of an irrationally mixable dimension group with cyclic rational subgroup is weakly mixing and  cannot be (strong) orbit equivalent to a Cantor minimal system with non-trivial spectrum.

Speaker: François Le Maître 

Title: L^1 full groups
We introduce a measurable analogue of the small topological full groups [[phi]], which we call L^1 full groups. These are Polish groups whose properties remain to be explored, but we will explain how in the case of Z actions one can show the following reconstruction theorem: if S and T are two ergodic transformations, their L^1 full groups are abstractly isomorphic iff S and T are flip conjugate.

Speaker: Nicolás Matte Bon

Random walks on topological full groups

A finitely generated group is sad to have the Liouville property if there
are no non-constant bounded harmonic functions on its Cayley graph. This
is an intermediate property betweeen subexponential growth and
amenability, closely related to random walks. I will show that it holds
for topological full group of a subshift with very slow word-complexity.
This provides the first examples of finitely generated simple groups with
the Liouville property, and it leads to estimates for the Folner function
of the topological full group of a class of minimal subshifts.

Speaker: Kostyantyn Medynets

Applications of Topological Orbit Equivalence to Representation Theory of Transformation Groups.

Studying representation theory of the infinite symmetric group S(N), Vershik noticed that "almost every"  character "f" of S(N) originated from an  ergodic action of  S(N) on  a measure space (X,mu) by the formula f(g) = mu(FixedPoints(g)).  Having established that every character for a group in question has such a dynamical   interpretation, one can  use topological dynamics techniques to classify ergodic measures and, as a result, to describe all group characters.  In view of the GNS-construction, the classification of characters is equivalent to the classification of II_1 (in the sense of Murray-von Neumann) representations of the group.  
We will reexamine  some older results on the classification of characters based off this new dynamics perspective and discuss the structure of  characters for the Higman-Thompson groups and full groups of Bratteli diagrams.  This talk is based on joint work with Artem Dudko.

Speaker: Julien Melleray

Title: Full groups and descriptive set theory

(Joint work with T. Ibarlucia) I will discuss some questions that someone interested in Polish groups might want to ask about full groups of minimal homeomorphisms. I will try to motivate these questions and promote the study of closures of full groups as an interesting direction of research.

Speaker: Samuel Petite

Title: Eigenvalues and strong orbit equivalence

In a join work with M.I. Cortez and F. Durand,we  give conditions on the subgroups of the circle to be realized as the subgroups of eigenvalues of minimal Cantor systems  belonging to a determined strong orbit equivalence class. Actually, the additive group of continuous eigenvalues  $E(X,T)$ of the minimal Cantor system $(X,T)$ is a subgroup of the  intersection $I(X,T)$ of all the images of the dimension group by its traces. We show, whenever the infinitesimal subgroup of the dimension group associated to $(X,T)$ is trivial,  the quotient group $I(X,T)/E(X,T)$ is torsion free. We give examples with non trivial infinitesimal subgroups where this property fails.  We also provide some realization results.

Speaker: Roman Sasyk

Title: Orbit equivalence and von Neumann algebras 

The purpose of this talk is to explain some of the connections between orbit equivalence, von Neumann algebras and set theory.

Speaker: Hisatoshi Yuasa

Uniform sets for infinite measure-preserving systems

Topological models of an ergodic automorphism of a Lebesgue probability space have been developed by for example [2, 3, 1, 5, 4]. This talk shows that any ergodic automorphism of an infinite Lebesgue space has a topological model acting on a locally compact, non-compact, metric space and having a unique, up to scaling, invariant Radon measure. A proof follows ideas due to [5].

1. G. Hansel and J. P. Raoult, Ergodicity, uniformity and unique ergodicity, Indiana Univ. Math. J. 23 (1973), 221–237.
2. R. I. Jewett, The prevalence of uniquely ergodic systems, J. Math. Mech. 19 (1970), 717–729.
3. W. Krieger, On unique ergodicity, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, vol. 2, University of California Press, Berkeley, Calif, 1972, pp. 327–346.
4. N. S. Ormes, Strong orbit realization for minimal homeomorphisms, J. Anal. Math. 71 (1997),103–133.
5. B. Weiss, Strictly ergodic models for dynamical systems, Bull. Amer. Math. Soc. (N.S.) 13 (1985), 143–146.