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.

We explain the recently introduced concept of fair measures, and how they can be used to understand randomly-chosen backward orbits of a dynamical system. We then extend the concept to countable-to-one maps and illustrate how to find these measures for a few examples of interval and dendrite maps.

J Ize conjectured that for any absolutely irreducible representation of a compact Lie group G on a finite dimensional real vectorspace there exists an isotropy subgroup which has an odd dimensional fixed point space. If it were true it had immediate consequences in equivariant bifurcation. Lauterbach & Matthews showed that this is not the case. Their findings of three infinite families of finite groups were supplemented by extensive computer analysis showing a very difficult zoo of groups acting on R4. In this talk we will give a complete list of counter examples in R4.

The concept of chaos is widely used in the field of Dynamical Systems, and several approaches which aim to establish the presence of chaotic dynamics have been developed in the literature. At this juncture, a prototypical example comes from the geometric structure associated with the Smale’s horseshoe, cf. [4]. In recent years, several different approaches have been proposed to extend this classical geometry in a topological direction. This way, the so-called concept of “topological horseshoes” was introduced in [2]. 

The concept of chaos is widely used in the field of Dynamical Systems, and several approaches which aim to establish the presence of chaotic dynamics have been developed in the literature. At this juncture, a prototypical example comes from the geometric structure associated with the Smale’s horseshoe, cf. [4]. In recent years, several different approaches have been proposed to extend this classical geometry in a topological direction. This way, the so-called concept of “topological horseshoes” was introduced in [2].

We study two variations of Bowen's definitions of topological entropy based on separated and spanning sets which can be applied to the study of discontinuous semiflows on compact metric spaces. We prove that these definitions reduce to Bowen's ones in the case of continuous semiflows. As a second result, we prove that our entropies give a lower bound for the τ-entropy defined by Alves, Carvalho and Vásquez (2015).