5 - Ergodic theory and linear dynamics

Summary

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 Poincaré 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 Tⁿ(x) ∈ A for infinitely many n ∈ N.

Now the central concept in linear dynamics is not recurrence but transitivity.