Functional analytic approach to dynamical systems I

I will discuss the study of the statistical properties of dynamicalsystems from a functional analytic point of view.Lecture 1: The basic idea: expanding maps. Synopsis: I will discusshow to study statistical properties and limit theorems for the simplecase of smooth expanding maps.Lecture 2: Hyperbolic systems and anisotropic Banach spaces. Synopsis:I will discuss how to upgrade the strategies put forward in theprevious lecture to the case of hyperbolic dynamics. To do so it isnecessary to introduce unconventional functional spaces. Such spacescame in various form, I will discuss the so called “geometric spaces”.Lecture 3: An application to the case of parabolic dynamics. Synopsis:I will discuss how parabolic dynamics can be treated viarenormalization techniques. In the case in which the renormalisingdynamics is hyperbolic, the results discussed in the previous lecturescan be used to obtain results on statistical properties of theparabolic system.