Embedding Bratteli-Vershik systems in cellular automata

TL;DR

This paper presents a method to embed Bratteli-Vershik systems, which describe various dynamical systems, into cellular automata by representing them as two-dimensional subshifts of finite type and then recoding as spacetime diagrams.

Contribution

The authors introduce a novel technique to embed a broad class of adic systems into cellular automata using combinatorial and symbolic dynamics methods.

Findings

Many odometers, Toeplitz systems, and substitution systems can be embedded in 1D cellular automata.

The embedding preserves the dynamical structure of the original systems.

The approach provides a new way to analyze complex systems via cellular automata.

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 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.