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

Abstract

We present a graph-theoretic model for dynamical systems $(X,σ)$ given by a surjective local homeomorphism $σ$ 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 $(X,σ)$ 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 $(X,σ)$, and we use the above approximation of $(X,σ)$ to write these algebras as colimits of the associated algebras of the corresponding generalized finite shifts, which we call generalized finite shift algebras.

Figures (12)

• Figure 1: The first three levels of an $l$-diagram.
• Figure 2: Construction of the edges of $(F,D)$.
• Figure 5: Construction of the map $\sigma$.
• Figure 6: The generalized finite shift $(E,C)$.
• Figure 7: Relation between $(\overline{e},e)$ and $\overline{e}'$ via red edges.

Theorems & Definitions (148)

• Theorem 1.1: Theorem \ref{['theorem-correspondence']}
• Theorem 1.2: Theorem \ref{['theorem-equivalence.homeo']}
• Theorem 1.3: Theorem \ref{['theorem-inverse.limit']}
• Theorem 1.4: Theorem \ref{['theorem-combined.FINAL.RESULT']}
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Definition 3.1