In this talk we will investigate non-asymptotic stability properties of simple, robust heteroclinic cycles for an ode \dot x = f(x), that are contained in a three-dimensional subspace of R^4. Such cycles are referred to as type B. There are four qualitatively distinct cycles of this type in R^4,...

In this talk, we study some global aspects of the bifurcation of an equivariant family of volume-contracting vector fields on the three-dimensional unitary sphere. The vector fields have two saddle-foci of different Morse index, connected in a cycle that is structurally stable within a class of...

We consider the problem of invisibility for bodies with mirror surface within the scope of geometrical optics. The problem amounts to studying billiards in the exterior of bounded regions. Examples of bodies invisible in 1, 2, and 3 directions and bodies invisible from 1 and 2 points are provided...

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The study of the long-time behavior of solutions to nonautonomous differential equations from the point of view of a dynamical system can be based on the notion of a pullback attractor, which plays a similar role as the global attractor in autonomous dynamical systems. In this lecture I will...

We consider systems for which there exists decay of correlations against $L^1$ observables. Examples include expanding and piecewise expanding systems. For such systems we consider rare events consisting on the entrance into very small neighbourhoods of some chosen points $z$ on the phase space....

We consider discrete time dynamical systems and show the link between Hitting Time Statistics (the distribution of the first time points land in asymptotically small sets) and Extreme Value Theory (distribution properties of the partial maximum of stochastic processes). This relation allows to...

A conjecture of Victor Ivrii (1980) says that in every billiard with smooth boundary the set of periodic orbits has measure zero. This conjecture is closely related to a conjecture of Hermann Weyl (1911) from the spectral theory. The particular case of Ivrii’s conjecture for triangular orbits was...

We consider a partially hyperbolic set K on a Riemannian manifold M whose tangent space splits as T_K M = E^{cu} ⊕ E^s, for which the center-unstable direction E^{cu} expands non-uniformly on some local unstable disk. We prove that the (stretched) expo- nential decay of recurrence time can...

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