Embedding Bratteli-Vershik systems in cellular automata

Details Embedding Bratteli-Vershik systems in cellular automata Embedding Bratteli-Vershik systems in cellular automata

2007-10-19

arxiv.org/abs/0710.3608v1

Mathematics

Dynamical Systems

12 pages, 3 figures

Scientific paper

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 F; in this way X is `embedded' in F (i.e. X is conjugate to a subsystem of F). With this technique, we can embed many odometers, Toeplitz systems, and constant-length substitution systems in one-dimensional cellular automata.

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