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

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 $C^*$ 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.