6 - Combinatorics on Bratteli diagrams and dynamical systems

Summary

The aim of this chapter is to show how Bratteli diagrams are used to study topological dynamical systems. We illustrate their wide range of applications through classical notions: invariant measures, entropy, expansivity, representation theorems, strong orbit equivalence, eigenvalues of the Koopman operator.

Introduction

In 1972 O. Bratteli (Bratteli 1972) introduced special infinite graphs – subsequently called Bratteli diagrams – which conveniently encoded the successive embeddings of an ascending sequence (A_n)_n≥0 of finite-dimensional semi-simple algebras over ℂ (‘multi-matrix algebras’). The sequence (A_n)_n≥0 determines a so-called approximately finite-dimensional (AF) C* -algebra. Bratteli proved that the equivalence relation on Bratteli diagrams generated by the operation of telescoping is a complete isomorphism invariant for AF-algebras.

From a different direction came the extremely fruitful idea of A. M. Vershik (Vershik 1985) to associate dynamics (adic transformations) with Bratteli diagrams (Markov compacta) by introducing a lexicographic ordering on the infinite paths of the diagram. By a careful refining of Vershik's construction, R. H. Herman, I. F. Putnam and C. F. Skau (Herman, Putnam, and Skau 1992) succeeded in showing that every minimal Cantor dynamical system is isomorphic to a Bratteli–Vershik dynamical system.

This chapter will give the details of this isomorphism and present some developments.

In this chapter all the dynamical systems (X, T) we consider are such that T is a homeomorphism. We thus work with the two-sided orbit of x ϵ X, that is, {Tⁿ x ∣ n ϵ ℤ}.