See attached file.

We discuss some recent and ongoing works on the dynamics of flows with various expansive measures. In particular, we present a measurable version of the Smale’s spectral decomposition theorem for flows. More precisely, we prove that if a flow φ on a compact metric space X is invariantly measure expanding on its chain recurrent setCR(φ) and has the invariantly measure shadowing property on CR(φ) then φ has the spectral decomposition, i.e. the nonwandering set Ω(φ) is decomposed by a disjoint union of finitely many invariant and closed sets on which φ is topologically transitive.

We discuss some recent and ongoing works on the dynamics of flows with various expansive measures. In particular, we present a measurable version of the Smale’s spectral decomposition theorem for flows. More precisely, we prove that if a flow φ on a compact metric space X is invariantly measure expanding on its chain recurrent setCR(φ) and has the invariantly measure shadowing property on CR(φ) then φ has the spectral decomposition, i.e. the nonwandering set Ω(φ) is decomposed by a disjoint union of finitely many invariant and closed sets on which φ is topologically transitive.

We consider stochastic processes arising from dynamical systems by evaluating an observable function along the orbits of the system. The Extremal Index, which is responsible for the appearance of clusters of exceedances, usually coincides with the reciprocal of the mean of the limiting cluster size distribution. In this talk, we show how to build dynamically generated stochastic processes with an Extremal Index for which that relation does not hold.

Discutiremos a dinâmica de grupos de difeomorfismos (analíticos) do círculo que são (localmente) não discretos. Uma técnica já bem estabelecida de aproximação desses grupos por campos de vectores locais permite uma descrição bastante detalhada das dinâmicas (extremamente caóticas) correspondentes.

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.