# Sistemas Dinámicos y Teoría Ergódica

Organizadores: Jairo Bochi (jairo.bochi@mat.uc.cl), Katrin Gelfert (gelfert@im.ufrj.br), Rafael Potrie (rpotrie@cmat.edu.uy)

• #### Jueves 16

15:00 - 15:45

#### Polynomial decay of correlations of geodesic flows on some nonpositively curved surfaces

Yuri Lima (Universidade Federal do Ceará, Brasil)

We consider a class of nonpositively curved surfaces and show that their geodesic flows have polynomial decay of correlations. This is a joint work with Carlos Matheus and Ian Melbourne.

15:45 - 16:30

#### Actions of abelian-by-cyclic groups on surfaces

I'll discuss some results and open questions about global rigidity of actions of certain solvable groups (abelian by cyclic) on two dimensional manifolds. Joint work with Jinxin Xue.

16:45 - 17:30

#### On tilings, amenable equivalence relations and foliated spaces

Matilde Martínez (Universidad de la República, Uruguay)

I will describe a family of foliated spaces constructed from tllings on Lie groups. They provide a negative answer to the following question by G.Hector: are leaves of a compact foliated space always quasi-isometric to Cayley graphs? Their construction was motivated by a profound conjecture of Giordano, Putnam and Skau on the classification, up to orbit equivalence, of actions of countable amenable groups on the Cantor set. I will briefly explain how these examples relate to the GPS conjecture. This is joint work with Fernando Alcalde Cuesta and Álvaro Lozano Rojo.

17:30 - 18:15

#### Conjugacy classes of big mapping class groups

Ferrán Valdez (Universidad Nacional Autónoma de México, México)

A surface $$S$$ is big if its fundamental group is not finitely generated. To each big surface one can associate its mapping class group, $$\mathrm{Map}(S)$$, which is $$\mathrm{Homeo}(S)$$ mod isotopy. This is a Polish group for the compact-open topology. In this talk we study the action of $$\mathrm{Map}(S)$$ on itself by conjugacy and characterize when this action has a dense or co-meager orbit. This is a joint work with Jesus Hernández Hernández, Michael Hrusak, Israel Morales, Anja Randecker and Manuel Sedano (arxiv.org/abs/2105.11282v2).

• #### Viernes 17

15:00 - 15:45

#### Multiplicative actions and applications

Sebastián Donoso (Universidad de Chile, Chile)

In this talk, I will discuss recurrence problems for actions of the multiplicative semigroup of integers. Answers to these problems have consequences in number theory and combinatorics, such as understanding whether Pythagorean trios are partition regular. I will present in general terms the questions, strategies from dynamics to address them and mention some recent results we obtained. This is joint work with Anh Le, Joel Moreira, and Wenbo Sun.

15:45 - 16:30

#### Continuity of center Lyapunov exponents.

Karina Marín (Universidade Federal de Minas Gerais, Brasil)

The continuity of Lyapunov exponents has been extensively studied in the context of linear cocycles. However, there are few theorems that provide information for the case of diffeomorphisms. In this talk, we will review some of the known results and explain the main difficulties that appear when trying to adapt the usual techniques to the study of center Lyapunov exponents of partially hyperbolic diffeomorphisms.

16:45 - 17:30

#### Lyapunov exponents of hyperbolic and partially hyperbolic diffeomorphisms

If $$f$$ is a diffeomorphism on a compact $$d$$-dimensional manifold $$M$$ preserving the Lebesgue measure $$\mu$$, then Oseledets Theorem tells us that almost every point has $$d$$ Lyapunov exponents (possibly repeated): $$\lambda_1(f,x)\leq\lambda_2(f,x)\leq\dots\leq\lambda_d(f,x).$$ If furthermore $$\mu$$ is ergodic, then the Lyapunov exponents are independent of the point $$x$$ (a.e.). We are interested in understanding the map $$f\in Diff_{\mu}^r(M)\ \mapsto\ (\lambda_1(f),\lambda_2(f),\dots,\lambda_d(f)),\ r\geq 1.$$ In general this map may be very complicated. However, if we restrict our attention to the set of Anosov or partially hyperbolic diffeomorphisms, then we can understand this map better. I will present various results related to the regularity, rigidity and flexibility of the Lyapunov exponents in this setting.
We review some recent results describing the behaviour of homeomorphisms of surfaces with zero topological entropy. Using mostly techniques from Brouwer theory, we show that the dynamics of such maps in the sphere is very restricted and in many ways similar to that of an integrable flow. We also show that many of these restrictions are still valid for $$2$$-torus homeomorphisms.