# Workshop on Combinatorics, Number Theory and Dynamical Systems

Semestre Temático – Dynamics Beyond Uniform Hyperbolicity

IMPA, Rio de Janeiro, de 19/08 até 23/08, 2013

**Financial Support: **

The workshop has a limited number of grants to cover local expenses of students and young researchers. Applications must be submitted until July 29th, and the result will be released on August 5th. All applicants must register at the workshop’s homepage.

Students must also submit a short curriculum vitae and a recommendation letter from the advisor or equivalent. The letter must be sent directly by the recommender to eventos@impa.br, with subject line “Workshop on Combinatorics, Number Teory and Dynamical Systems”.

## Abstracts

Abstract

## Certificates

## Invited Speakers

Boris Adamczewski (CNRS, France)

Pierre Arnoux (Université Aix-Marseille, France)

Artur Avila (IMPA, Brazil/CNRS, France)

Nilson Bernardes (UFRJ, Brazil)

Vitaly Bergelson (Ohio State University, USA)

Julien Cassaigne (CNRS, France)

Alex Eskin (University of Chicago, USA)

Sébastien Ferenczi (CNRS, France)

Albert Fisher (USP, Brazil)

Arek Goetz (San Francisco State University, USA)

Harald Helfgott (CNRS, France)

Yoshiharu Kohayakawa (USP, Brazil)

Elon Lindenstrauss (Hebrew University, Israel)

Ali Messaoudi (UNESP – S.J. do Rio Preto, Brazil)

János Pintz (Alfréd Rényi Institute of Mathematics, Hungary)

Miguel Walsh (Universidad de Buenos Aires, Argentina)

Barak Weiss (Tel Aviv University, Israel)

Maté Wierdl (University of Memphis, USA)

## Minicourses

In the two weeks prior to the workshop, from August 5th to August 16th, there will be two minicourses. The purpose of the minicourses is to introduce Ph.D. students and researchers to two prominent areas of research in dynamical systems and ergodic theory, namely Ratner’s theory and Ergodic Ramsey Theory. The idea is to develop the basics and discuss the main results in these areas.

The lectures of the workshop will overlap many of the topics discussed in the minicourses, thus we encourage those interested in the workshop to attend the minicourses. Below we list the tentative syllabus.

**Introduction to Ratner’s theory.**

**Speakers: **Alex Eskin and Yuri Lima

**Days:**

3ª (08/06),4ª(08/07), 5ª(08/08) e 6ª(08/09) – 15:30 – 17:00 – Aud. R. Mañé

3ª(08/13) e 5ª(08/15) – 15:30 – 17:00 – Aud. 3

**1.Dynamics on hyperbolic surfaces.**

-The hyperbolic plane: Möbius transformations, geodesics, geodesic and horocycle flows.

-The group SL(2,R): its linear and hyperbolic actions, algebraic description of geodesic and horocycle flows, ergodic theory of Anosov flows.

**2.Ratner’s theorems.**

-Homogeneous spaces: group actions, Lie groups, Haar measures, quotient spaces and lattices.

-Unipotent actions: Lie algebras, the adjoint representation.

-Ratner’s theorems: orbit closure theorem, measure classification theorem, and equidistribution theorem.

**3.Horocycle flows on hyperbolic surfaces.**

-More on SL(2,R): KAN and Cartan decompositions, Haar measure.

-Hedlund’s theorem.

-Mautner phenomenon: representations, Mautner lemma, Mautner phenomenon on SL(2,R).

-Howe-Moore’s theorem.

-Furstenberg’s theorem.

-Dani-Smillie’s theorem.

**4.Oppenheim’s conjecture.**

-The moduli space SL(n,R)/SL(n,Z): Mahler compactness criterium.

-Raghunathan’s observation.

-Proof of Oppenheim’s conjecture.

**Introduction to Ergodic Ramsey Theory.**

**Speaker: **Vitaly Bergelson

**Title:** Ergodic Ramsey theory

**Days:**

2ª (08/12), 3ª(08/13), 4ª(08/14) e 6ª(08/16) – 17:30 – 19:30 – Aud. 3

1.Three main principles of ergodic Ramsey theory.

2.Many facets of Poincaré recurrence theorem.

3.Uniform distribution and van der Corput sets.

4.Multiple recurrence, Szemeredi’s theorem and its extensions.

5.Topological algebra in the Stone-Cech compactifications and Ramsey theory.

6.Mild mixing, IP-sets and p-limits.

7.The nilpotent connection.

8.Ergodic Ramsey Theory and amenable groups.

## Picture of the Group

## Poster Session

Danilo Caprio (UNESP)

Fibonacci Adding Machine

Patricia Cirilo (UNESP)

Law of large numbers for certain cylinder flows

Pierre-Antoine Guihéneuf (Paris Sud 11)

What is the effect of spatial discretizations on dynamical systems?

Carlos Alberto Lima (USP-S.CARLOS)

Hausdorff Dimension Of Quasi-Arcs

Maurizio Monge (UNIPI)

On the failure of Kronecker’s density theorem for powers of an algebraic number

Tatiana Rodrigues (UNESP)

A class of Cubic Rauzy Fractal

Manuel Stadlbauer (UFBA)

A Ruelle theorem for transient random walks

**Poster Submission**

Posters should be sent to eventos@impa.br using __Poster – Workshop in Combinatorics__ as subject. The standard adopted for posters is size A0 vertical.

Deadline for submission of posters: June 29,2013.

## Program

## Registered Participants

## Scientific and Organizing Committee

Yuri Lima (Weizmann, Israel)

Christian Mauduit (CNRS/Université Aix-Marseille, France)

Carlos Gustavo Moreira (IMPA, Brazil)

Enrique Pujals (IMPA, Brazil)

Carlos Matheus Santos (CNRS, France)

Marcelo Viana (IMPA, Brazil)

Jean-Christophe Yoccoz (Collège de France, France)

## Video

**Postal Address: **Instituto Nacional de Matemática Pura e Aplicada

Estrada Dona Castorina 110, Jardim Botânico

Rio de Janeiro, RJ, CEP 22460-320, Brasil

E-mail: eventos@impa.br