Symbolic dynamics and the category of graphs (Q2884471)

Let \textbf{Gph} denote the category of directed graphs. In the paper \textbf{Gph} is investigated as a framework for analyzing symbolic dynamics of walks. By a walk in a directed graph \(X\) is meant a graph morphism from \(\mathbf N\) to \(X\), where \(\mathbf N\) is a special type of a Cayley graph, i.e., the graph which, for each nonnegative integer \(n\), has a node \(n\) and an arc from \(n\) to \(n+1\). On the category of directed graphs, a Quillen model structure is considered. For this structure the weak equivalences are those graph morphisms which induce bijections on the set of walks. The resulting homotopy category is determined. A ''finite-level'' homotopy category which respects the natural topology on the set of walks is also introduced. To each graph a basal graph, well defined up to isomorphism, is associated. It is shown that the basal graph is a homotopy invariant for the model structure, and that it is a finer invariant than the zeta series of a finite graph. It is also shown that, for finite walkable graphs, if \(B\) is a basal and separated then the walk spaces for \(X\) and \(B\) are topologically conjugate if and only if \(X\) and \(B\) are homotopically equivalent for the model structure considered. Some open problems are posed.