# School on orbit equivalence and related fields

Dec 1-5, 2014

Universidad de Santiago de Chile, Santiago, CHILE.

The School on orbit equivalence and related fields is a meeting that will focus on orbit equivalence and different aspects from Topological Dynamics and Ergodic Theory.

The school will be oriented to graduate students and researchers, and the program will consist of mini-courses and talks.

It will be held from the 1th of december to the 5th of december 2014, at the Departamento de Matemática y Ciencia de la Computación of the Universidad de Santiago de Chile.

**Organizing Committee:** José Aliste (Universidad Andrés Bello),

Daniel Coronel (Universidad Andrés Bello),

María Isabel Cortez (Universidad de Santiago de Chile),

Michael Schraudner (Universidad de Chile).

**Remark: **During the two weeks after the School, two other conferences will take place in Santiago: «Information and Randomness 2014» (from December 8th to 12th) and » Symbolic dynamics on (finitely presented) groups» (from December 15th to 19th).

**Damien Gaboriau** (École Normale Supérieure de Lyon)**Title:** Measurable Group Theory.**Abstract. **The goal of this series of lectures is to present an overview of the theory of orbit equivalence of measure preserving actions of countable groups, with a particular focus on the free groups. I will give several examples and explain some tools and invariants such as the theory of cost, the fundamental groups, measure equivalence of groups… I will present the framework for a measurable solution to the problem of von Neumann about non-amenability vs containment of a free subgroup.

**Thierry Giordano** (University of Ottawa), **Ian F. Putnam **(University of Victoria), **Christian Skau** (Norwegian University of Science and Technology)**Title**: Topological orbit equivalence and minimal dynamics on the Cantor set. **Abstract:** We will give an overview of minimal dynamical systems on the Cantor set with the aim of describing a complete invariant for orbit equivalence of actions of finitely generated abelian groups. Of course, this builds heavily on the similar program in ergodic theory initiated by Henry Dye, but we will focus on the aspects which are different in the topological setting. Many of the tools and invariants have their origin in C*-algebra theory, but our approach will be completely dynamical.

**Kate Juschenko** (Northwestern University) **Title: **Amenability and algebraic properties of the full topological groups.

**Abstract**. in four lectures we will present as complete as possible proof of the fact that the commutator subgroup of minimal subshift is simple, finitely generated, amenanle and there are uncountably many of such.

**David Kerr** (Texas A&M University)**Title: **Sofic entropy.

**Abstract. **I will present the basic theory and major developments in the subject of

sofic entropy since its inception a little more than five years ago in the breakthrough

work of Lewis Bowen. I will also discuss how these ideas relate to the classical

Kolmogorov-Sinai picture of dynamical entropy, which applies most generally to actions

of amenable groups. Topics will include Gottschalk´s surjunctivity conjecture,

Bernoulli actions, algebraic actions, and the f-invariant.

**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.

**Speaker:** 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.

**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

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

**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.

- Mahsa Allahbakhshi, CMM Universidad de Chile
- Fernando Alcalde, Universidad de Santiago de Compostela
- José Aliste-Prieto, Universidad Andrés Bello
- Drew Ash, University of Denver
- Jean-Baptiste Aujogue, Universidad de Santiago
- Nicolas Bedaride, Université Aix Marseille
- Matthieu Calvez, Universidad de Santiago
- James Campbell, University of Memphis
- Paulina Cecchi, Universidad de Chile
- Nishant Chandgotia, University of British Columbia
- Daniel Coronel, Universidad Andrés Bello
- María Isabel Cortez, Universidad de Santiago
- Yves de Cornulier, Université Paris Sud
- Martin Delacourt, CMM Universidad de Chile
- Alexander Frank, CMM Universidad de Chile
- Felipe Fresno, Universidad de Santiago
- Damien Gaboriau, École Normale Supérieure de Lyon
- Thierry Giordano, University of Ottawa
- Ricardo Gómez, Universidad Nacional Autónoma de México
- Daniel Gonçalves, Universidad Federal de Santa Catarina
- Benjamin Hellouin, Université Aix Marseille
- Maryam Hosseini, University of Ottawa
- Lison Jacoboni, Université Paris Sud
- Kate Juschenko, Northwestern University
- David Kerr, Texas A&M University
- Martha Lacka, Jagiellonian University in Krakow
- François Le Maître, Université de Louvain
- Douglas Lind, University of Washington
- Nicolas Matte Bon, Université Paris Sud
- Hugo Maturana, Universidad de Santiago
- Kostyantyn Medynets, United States Naval Academy
- Julien Melleray, Université Claude Bernard Lyon 1
- Andrés Navas, Universidad de Santiago
- Marcelo Paredes, Universidad de Buenos Aires
- José Pérez, Universidad de Santiago
- Samuel Petite, Université de Picardie Jules Verne
- Facundo Poggi, Universidad de Buenos Aires
- Ian Putnam, University of Victoria
- Igraine Quiroz, Universidad de Santiago
- Cristóbal Rivas, Universidad de Santiago
- Ville Salo, CMM Universidad de Chile
- Roman Sasyk, Universidad de Buenos Aires
- Michael Schraudner, Universidad de Chile
- Jacek Serafin, Wrocław University of Technology
- Christian Skau, Norwegian University of Science and Technology
- Hong Soonjo, CMM Universidad de Chile
- Mahdi Teymuri, Universidad de Santiago
- Siming Tu, CMM Universidad de Chile
- Andrea Vera, Universidad de Santiago
- Reem Yasawi, Trent University
- Hisatoshi Yuasa, Osaka Kyoiku University

For any information regarding the school please send an email to María Isabel Cortez to the following address:

maria.cortez (at) usach (dot) cl

Registration form is already closed. If you wish to attend the school, please contact María Isabel Cortez directly at maria.cortez@usach.cl

**Center of Dynamical Systems and Related Fields (ACT project 1103 PIA-CONICYT)**