2 - SUBSTITUTION SUBSHIFTS AND BRATTELI DIAGRAMS

Summary

Bernard HOST

Équipe d'Analyse et de Mathématiques Appliquées

Université de Marne la Vallée

2 rue de la Butte Verte

93166 Noisy le Grand cedex

France

The goal of these lectures is to introduce, through the example of substitution dynamical systems, some of the basic concepts of topological dynamics: minimality, unique ergodicity, Kakutani-Rohlin partitions, …

Moreover, we shall present here the less classical notions of Bratteli diagrams and Bratteli–Vershik systems. It became clear in the last years that these objects provide a very fruitful link between topological dynamics and the theory of C*-algebras. In particular, an algebraic invariant of C*-algebras, the so-called “dimension group”, has been proved to have a dynamical interpretation, and can now be used for the classification of dynamical systems.

Since we cannot completely develop this theory in the available space, we shall only explain how to construct a Bratteli–Vershik model in the case of substitution dynamical systems.

Subshifts

Notation: Words, Sequences, Morphisms

Words. An alphabet is a finite set of symbols called letters. If A is an alphabet, a word of A is a finite (non-empty) sequence of letters, and A⁺ denotes the set of words. For u = u₁u₂ … u_n ∈ A⁺, ∣u∣ = n is called the length of u. For each letter a we denote by ∣u∣_a the number of occurrences of a in u and the vector (∣u∣_a; a ∈ A) is sometimes called the composition vector of u. A* consists of A⁺ and the empty word ∅ of length 0.