A correspondence between surjective local homeomorphisms and a family of separated graphs

Abstract

We present a graph-theoretic model for dynamical systems $(X,\sigma )$ given by a surjective local homeomorphism $\sigma$ on a totally disconnected compact metrizable space $X$. In order to make the dynamics appear explicitly in the graph, we use two-colored Bratteli separated graphs as the graphs used to encode the information. In fact, our construction gives a bijective correspondence between such dynamical systems and a subclass of separated graphs which we call $l$-diagrams. This construction generalizes the well-known shifts of finite type, and leads naturally to the definition of a generalized finite shift. It turns out that any dynamical system $(X,\sigma )$ of our interest is the inverse limit of a sequence of generalized finite shifts. We also present a detailed study of the corresponding Steinberg and $C^{\ast }$ algebras associated with the dynamical system $(X,\sigma )$, and we use the above approximation of $(X,\sigma )$ to write these algebras as colimits of the associated algebras of the corresponding generalized finite shifts, which we call generalized finite shift algebras.

66 pages. This version matches the published version at "Discrete and Continuous Dynamical Systems"

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