Dimension Groups and Dynamical Systems

Abstract

The text follows, at least in an initial version, closely the notes of Bernard Host Host (1995), trying to develop more explicitly some arguments. We have also chosen to follow a slightly different presentation, not assuming systematically that the dynamical systems are minimal. We wish to thank Brian Marcus and Mike Boyle for fruitful discussions during the preparation of this text. We are also grateful to Paulina Cecchi, Francesco Dolce, Pavel Heller, Christophe Reutenauer and Revekka Kyriakoglou for reading the manuscript and founding many errors. Special thanks are due to Soren Eilers for reading closely Chapter 10 and to Christian for his careful reading of most chapters.Proposition 2.1.4 A topological dynamical system is minimal if and only if the positive orbit of every point is dense in X. Example 2.5.1 The Fibonacci shift (see Example 2.4.1) is Sturmian (see Exercise 2.32).As one might expect in view of its minimal word complexity, any Sturmian shift is minimal, moreover it is uniquely ergodic.