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.

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.

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.

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).

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.

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.

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.

Some of the results presented are joint with C. Vasquez, F. Valenzuela, J. Yang and P. Carrasco.

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.