Ergodic Theory and Dynamical Systems

Research Article

Finite-rank Bratteli–Vershik diagrams are expansive

TOMASZ DOWNAROWICZa1 and ALEJANDRO MAASSa2

a1 Institute of Mathematics, Technical University, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland (email: downar@im.pwr.wroc.pl)

a2 Department of Mathematical Engineering and Center of Mathematical Modeling, University of Chile, Av. Blanco Encalada 2120, 5to piso, Santiago, Chile (email: amaass@dim.uchile.cl)

Abstract

The representation of Cantor minimal systems by Bratteli–Vershik diagrams has been extensively used to study particular aspects of their dynamics. A main role has been played by the symbolic factors induced by the way vertices of a fixed level of the diagram are visited by the dynamics. The main result of this paper states that Cantor minimal systems that can be represented by Bratteli–Vershik diagrams with a uniformly bounded number of vertices at each level (called finite-rank systems) are either expansive or topologically conjugate to an odometer. More precisely, when expansive, they are topologically conjugate to one of their symbolic factors.

(Received May 10 2006)

(Revised May 02 2007)