On the asymptotic behaviour of piecewise expanding transformations

Abstract: 

In this talk, I will explain how the long-term behaviour of discrete dynamical systems can be studied with the help of the Frobenius-Perron operator. This operator describes how distributions change under the action of the transformation governing the dynamical system. For a large class of transformations - namely, those that are piecewise C^2 and expanding - a substantial body of theory guarantees the existence of well-defined asymptotic distributions and gives us reliable methods for computing them. When the transformations are less regular than C^2, we must look to newer techniques. I will introduce one such technique recently proposed by Carlangelo Liverani, and show that this construction allows us to study less regular transformations within the same framework developed for the classical C^2 case.