Given a heteroclinic network, there is an associated graph such that the vertices of the graph correspond to the equilibria of the network and an edge corresponds to a connection between equilibria. Classification of ac-networks is carried out by describing all possible types of asso- ciated graphs.

The embedding problems aim to describe when a certain map f (homeomorphism, diffeomorphism, etc) 

on a topological space X can be embedded in a flow with the same regularity.  There are fundamental contributions  

to the embedding problem for both the setting of homeomorphisms in low dimension (one and two) and diffeomorphisms 

dated from the fifties and the sixties, due to Anderson, Andrea, Foland, Fort, Palis, Utz and Zdun among others. 

There are few explicit examples in the literature of non-autonomous vector fields exhibiting complex dynamics that may be proven analytically. In this talk, we present a careful analysis of the rich dynamics generated by a family of periodic perturbations of a weakly attracting robust heteroclinic network defined on the two-sphere.

We present one–parameter families of differentiable planar vector fields for which the infinity reverses its stability as the parameter goes through zero. These vector fields are defined on the complement of some compact ball centered at the origin and have isolated singularities. They may be considered as linear perturbations at infinity of a vector field with some spectral property, for instance, dissipativity. We also address the case concerning linear perturbations of planar systems with a global period annulus.

In this seminar we prove the statistical stability of Luzzatto-Viana maps. We show that for each parameter, the map has non-uniform expanding behavior and slow recurrence to the critical/singular set with exponential decay of the measure of the tail set. In this setting, we also obtain transitivity of the map, and so the uniqueness of the physical measure. This improves a result by Araújo, Luzzatto and Viana (2009), where they prove the existence of a finite number of the physical measures.

Nesta palestra discutimos a existência e unicidade de estados de equilíbrio para uma classe (grande) de produtos cruzados associados a potenciais naturais. Em particular, estes resultados podem-se aplicar aos exemplos clássicos não hiperbólicos de Abraham-Smale e Shub. Este e um trabalho em colaboração com Maria Carvalho.

A map f : [0, 1] → [0, 1] is a piecewise contraction if locally f contracts distance, i.e., if there exist 0 < λ < 1 and a partition of [0,1] into intervals I1,I2,...,In such that |f(x) − f(y)| ≤ λ|x − y| for all x, y ∈ Ii (1 ≤ i ≤ n). Piecewise contractions describe the dynamics of many systems such as traffic control systems, queueing systems, outer billiards and Cherry flows. Here I am interested in the symbolic dynamics of such maps. More precisely, we say that an infinite word i0 i1i2 . . .

In this seminar we present the normal form theory as a tool for the local study of the qualitative behavior of hamiltonian vector fields under the action of a linear Lie group Γ. We introduce an algebraic method which takes into account a given group S of matrices in terms of the linear part of the associated hamiltonian system. The normal form can inherit the symmetries of the original hamiltonian function if the changes of coordinates are equivariant under the action Γ. For this study, we use tools from the representation theory and invariant theory of linear Lie groups.

In the last three decades several works have been done on the stable ergodicity problem. Almost every work that exists was done in the partially hyperbolic setting. The only result about stable ergodicity outside the partially hyperbolic world was done by Tahzibi in 2004, where he gives examples of stably ergodic diffeomorphisms which are not partially hyperbolic. In this seminar I will talk about a criterium of stable ergodicity for diffeomorphisms with dominated splitting and some applications of this criterium. 

 In this talk I will introduce translated cone exchange transformations, a new family of piecewise isometries and renormalize its first return map to a subset of its partition. As a consequence I will show that the existence of an embedding of an interval exchange transformation into a map of this family implies the existence of infinitely many bounded invariant sets. Finally, I will prove the existence of infinitely many periodic islands, accumulating on the real line, as well as non-ergodicity of our family of maps close to the origin. This is joint work with Pedro Peres.