Os célebres resultados de S.Newhouse ([N]) mostram que a bifurcação de uma tangência homoclínica asociada a uma sela numa superfície gera tangências homoclínicas robustas (isto é, tangências homoclínicas que persistem por pequenas perturbações) associadas a um conjunto hiperbólico especial chamado ferradura espessa. Além disso, a continuação (hiperbólica) da sela inicial está contida nesse conjunto hiperbólico.

# Seminars

Let *X* be a set, *X'* be a disjoint copy of *X *and $\bar{X}\wedge\bar{X}=\{(x\wedge y): x,y\in X\cup X'\}$. We look at $\hat{X}=X\cup X'\cup(\bar{X}\wedge \bar{X})$ as a set of letters and consider the free semigroup $\hat{X}^+$ on the set $\hat{X}$. Auinger [1] constructed a model for the bifree locally inverse semigroup on *X* as a quotient semigroup of $\hat{X}^+$. This result enables us to talk about presentations $\langle X;R\rangle$ of locally inverse semigroups (LI-presentations) where $R\subseteq \hat{X}^+\times\hat{X}^+$.

In this seminar, we explore the chaotic set near a homoclinic cycle to a hyperbolic bifocus at which the vector field has negative divergence. If the invariant manifolds of the bifocus satisfy a non-degeneracy condition, a sequence of hyperbolic suspended horseshoes arises near the cycle, with one expanding and two contracting directions.

We shall discuss residual properties of groups and their interpretation in connection with the profinite completion of groups of geometric nature.

We present the result that, under a certain condition, free pro-$p$ products with procyclic amalgamation inherit from its free factors the property of each 2-generator pro-$p$ subgroup being free pro-$p$. This generalizes known pro-$p$ results, as well as some pro-$p$ analogues of classical results in Combinatorial Group Theory.

A nonhyperbolic ergodic measure is an ergodic invariant measure with one Lyapunov exponent equal zero. Gorodetski, Ilyashenko, Kleptsyn, and Nalsky constructed a nonhyperbolic ergodic measure for a skew product diffeomorphism of the three-dimensional torus. Inspired by this construction, Bonatti, Diaz and Gorodetski gave sufficient condi- tions for weak convergence of a sequence of measures supported on periodic orbits to an ergodic measure. A royal measure is a measure obtained through this scheme.

In this talk we intend to present the denition of mean topological dimension for a topological dynamical system. It is a topological invariant which was introduced by M. Gromov and exploited by some authors. This invariant enables one to assign a meaningful quantity to a dynamical system of infinite topological entropy. Also is possible to obtain a kind of variational principle as we intend to show.

The celebrated Birkhoff's ergodic theorem asserts that from a probabilistic viewpoint the times averages of "almost all" points converge to a space average. Motivated by the application of iterated function systems (IFS) to model central dynamics of partially hyperbolic diffeomorphisms, we will describe mild conditions that ensure that Birkhoff non-typical points form a Baire generic subset. If time permits we will provide some applicationsof this result in a partial hyperbolicity context. This is a ongoing joint work with my postdoctoral student G. Ferreira (UFMA).

For an arbitrary group G, it is shown that either the semigroup rank GrkS equals the group rank GrkG, or GrkS = GrkG +1. This is the starting point for the rest of the work, where the semigroup rank for diverse kinds of groups is analysed. The semigroup rank of any relatively free group is computed. For a finitely generated abelian group G, it is proven that GrkS = GrkG+1 if and only if G is torsion-free. In general, this is not true. Partial results are obtained in the nilpotent case.

Kiem proved that the moduli space of SL(2,C)-Higgs bundles over a smooth projective curve X admits a symplectic desingularization if and only if g(X) = 2. Based on work of Schedler and Bellamy, Tirelli reproduced this result to the case of GL(2, C)-Higgs bundles, proving as well that no symplectic desingularization of SL(n, C) and GL(n, C)-Higgs bundles exists besides these two cases where (n, g) = (2, 2).