Fermi Acceleration in anti-integrable limits of the standard map

We consider a dynamical system on the semi-infinite cylinder which models the high energy dynamics of a family of mechanical models. We provide conditions under which we ensure that the set of orbits undergoing Fermi acceleration has measure zero

Fast-slow partially hyperbolic systems versus Freidlin-Wentzell random systems

We consider a simple class of fast-slow partially hyperbolic dynamical systems and show that the (properly rescaled) behaviour of the slow variable is very close to a Friedlin--Wentzell type random system for times that are rather long, but much shorter than the metastability scale. Also, we show the possibility of a "sink" with all the Lyapunov exponents positive, a phenomenon that turns out to be related to the lack of absolutely continuity of the central foliation.Comment: To appear in Journal of Statistical Physic

The Martingale approach after Varadhan and Dolpogpyat

We present, in the simplest possible form, the so called {\em martingale problem} strategy to establish limit theorems. The presentation is specially adapted to problems arising in partially hyperbolic dynamical systems. We will discuss a simple partially hyperbolic example with fast-slow variables and use the martingale method to prove an averaging theorem and study fluctuations from the average. The emphasis is on ideas rather than on results. Also, no effort whatsoever is done to review the vast literature of the field