Dynamics without physical measures

Overview & Synopsis

In this thesis, we study the statistical properties of dynamical systems. More precisely, we study the asymptotic statistical behavior of typical orbits via the convergence of the sequence of empirical measures. We also study the stability of asymptotic behavior under the effect of small perturbations of the dynamics within a family of maps.This note has two parts. In the first part, we link the two subjects mentioned above. The second part is devoted to the proof of the existence of non-statistical dynamics in the space of rational maps of the Riemann sphere. A map is said to be non-statistical with respect to the reference measure, if there is a set of positive measures of points which have a divergent sequence of empirical measures.In the first part, we have developed an abstract setting which aims to under-stand what lies behind the existence of non-statistical maps in a given family of dynamics. In this part, we extend and formalize the concept of statistical instability in a general sense, independent of the statistical behavior of the dynamics, and we show how the abundance of statistically unstable maps, in a given family of dynam-ics, implies the existence of non-statistical maps in this family. We provide another application of this formalization, and also another example of non-statistical maps: non-statistical Anosov-Katok diffeomorphisms of the annulus.Keywords: Non-statistical dynamics, statistical instability, rational maps, Anosov-Katok diffeomorphisms