# Talks

**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**

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**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.

**
Speaker**: Yves Cornulier

**Title:**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

**Title:** On Orbit Equivalence of Cantor Minimal Systems and their Continuous Spectrum

*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

**Speaker:**Nicolás Matte Bon

**Title:**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

**Title:**Applications of Topological Orbit Equivalence to Representation Theory of Transformation Groups.

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

**Title:**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.