Absolutely Continuous Spectrum and Inverse Theory

Abstract

We assume that a complete dynamical system(Ω,F,ℙ, {θ _t ;t ∈ T} is fixed throughout the chapter. In other words, we assume that (Ω,F,ℙ) is a complete probability space and {θ _t ;t ∈ T} is an ergodic group of automorphisms of the measurable space (Ω,F) which leave invariant the measure ℙ. Except for the short last section, we shall not consider the multidimensional case and we concentrate on the cases T = ℝ and T = ℤ. We will add most of the time the assumption of ergodicity of the flow {θ _t ;t ∈ T} of transformations. All the potential functions considered in this chapter are assumed to be defined on such a dynamical system. They are even assumed to be of the form V(t,ω) = υ(θ _t ω) for some measurable function υ : Ω → ℝ whenever possible.