Embedding Bratteli-vershik Systems in Cellular Automata *

Many dynamical systems can be naturally represented as Bratteli-Vershik (or adic) systems, which provide an appealing combinatorial description of their dynamics. If an adic system X satisfies two technical conditions (focus and bounded width) then we show how to represent X using a two-dimensional subshift of finite type Y ; each ‘row’ in a Y -admissible configuration corresponds to an infinite path in the Bratteli diagram of X, and the vertical shift on Y corresponds to the ‘successor’ map of X. Any Y -admissible configuration can then be recoded as the spacetime diagram of a one-dimensional cellular automaton Φ; in this way X is embedded in Φ (i.e. X is conjugate to a subsystem of Φ). With this technique, we can embed many odometers, Toeplitz systems, and constant-length substitution systems in one-dimensional cellular automata.