Étienne Ghys (ENS – Lyon)

In dynamical systems one usually considers the dynamics of "typical diffeomorphisms". Of course, one of the very first questions is to define "typical"! Pioneers used Baire category: countable intersections of open and dense sets. Later, Kolmogorov suggested to use the concept which is called today "prevalence": some kind of substitute for the Lebesgue measure in infinite dimension. In this talk, I will begin by explaining the advantages and drawbacks of these two notions. Then, I will restrict myself to the 1 dimensional case and discuss the Malliavin-Shavgulidze measure on the group of diffeomorphisms of the circle, related to the Brownian motion. It will be a pleasant opportunity to advertise part of the PhD thesis of my latest student: Michele Triestino. One would like to understand the dynamics of almost all diffeomorphism of the circle, with respect to this Malliavin-Shavgulidze probability.