Ergodic theory and linear dynamics

Abstract

Introduction So far, we have obtained hypercyclic vectors either by a direct construction or by a Baire category argument. The aim of this chapter is to provide another way of doing so, using ergodic theory . This will link linear dynamics with measurable dynamics. We first recall some basic definitions from ergodic theory. The classical book of P. Walters [235] is a very readable introduction to that area. The first important concept is that of invariant measure. DEFINITION 5.1 Let ( X , B, μ) be a probability space. We say that a measurable map T : ( X , B, μ) → ( X , B, μ) is a measure-preserving transformation , or that μ is T- invariant , if μ ( T –1 ( A )) = μ( A ) for all A ∈ B. Measure-preserving transformations already have some important dynamical properties. In particular, the famous Poincare recurrence theorem asserts that if T : ( X , μ) → ( X , μ) is measure-preserving then, for any measurable set A such that μ( A ) > 0, almost every point x ∈ A is T - recurrent with respect to A , which means that T n ( x ) ∈ A for infinitely many n ∈ N. Now the central concept in linear dynamics is not recurrence but transitivity.