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Embedding Bratteli–Vershik systems in cellular automata

Published online by Cambridge University Press:  15 October 2009

MARCUS PIVATO
Affiliation:
Department of Mathematics, Trent University, 1600 West Bank Drive, Peterborough, Ontario, K9J 7B8, Canada (email: marcuspivato@trentu.ca, ryassawi@trentu.ca)
REEM YASSAWI
Affiliation:
Department of Mathematics, Trent University, 1600 West Bank Drive, Peterborough, Ontario, K9J 7B8, Canada (email: marcuspivato@trentu.ca, ryassawi@trentu.ca)

Abstract

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 is linearly recurrent, 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 space-time 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.

Type
Research Article
Copyright
Copyright © Cambridge University Press 2009

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