1   2008-08-28 Description

The course will focus on the measure theoretic and statistical properties of dynamical systems. We begin with the Poincare recurrence theorem and the Birkhoff ergodic theorem (which relates spatial averages to time averages) and some of their applications (including Borel’s theorem on normal numbers). We then look more carefully at the ergodic properties of invariant measures. The set of invariant measures is convex and its extremal points are ergodic measures. Then we will introduce the metric entropy (Kolmogorov entropy) as an average density of the information content of a coding and prove the equivalence to the Shannon entropy (Shannon-McMillan-Breiman theorem). Next we consider invariant measures whose densities are governed by the presence of a potential function. This allows us to introduce the pressure function and prove existence and uniqueness of the associated equilibrium states using thermodynamic formalism which includes the transfer operator which was originally introduced by Ising at the beginning of the 20th century. Its spectral properties allows to deduce statistical properties of a dynamical system and obtain the decay rate of correlation functions. We conclude the course with the dynamical zeta function, its analytic properties and an application similar to the prime number theorem which allows to find the limiting behaviour of the length spectrum of closed orbits. The theoretical aspects will be highlighted by examples and in particular their application to symbolic systems where they have a strong connection to coding theory.