Dynamical Systems
Binary De Bruijn graphs can be drawn (below, left) in such a way that they resemble objects from the theory of dynamical systems, such as the Lorenz attractor (below, right):
This analogy can be made rigorous: the n-dimensional m-symbol De Bruijn graph is a model of the Bernoulli map
The Bernoulli map (also called the 2x mod 1 map for m = 2) is an ergodic dynamical system, which can be understood to be a single shift of a m-adic number. The trajectories of this dynamical system correspond to walks in the De Bruijn graph, where the correspondence is given by mapping each real x in the interval [0,1) to the vertex corresponding to the first n digits in the base-m representation of x. Equivalently, walks in the De Bruijn graph correspond to trajectories in a one-sided subshift of finite type.
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