Symbolic dynamics and the category of graphs

Symbolic dynamics is partly the study of walks in a directed graph. By a walk, here we mean a morphism to the graph from the Cayley graph of the monoid of non-negative integers. Sets of these walks are also important in other areas, such as stochastic processes, automata, combinatorial group theory, $C^*$-algebras, etc. We put a Quillen model structure on the category of directed graphs, for which the weak equivalences are those graph morphisms which induce bijections on the set of walks. We determine the resulting homotopy category. We also introduce a “finite-level” homotopy category which respects the natural topology on the set of walks. To each graph we associate a basal graph, well defined up to isomorphism. We show that the basal graph is a homotopy invariant for our model structure, and that it is a finer invariant than the zeta series of a finite graph. We also show that, for finite walkable graphs, if $B$ is 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 our model structure.

💡 Research Summary

The paper develops a new categorical framework that connects symbolic dynamics, graph theory, and homotopy theory by focusing on walks in directed graphs. A walk is defined as a graph morphism from the Cayley graph of the monoid of non‑negative integers (ℕ) into a given directed graph G. In this setting a walk is simply an infinite sequence of vertices (v₀, v₁, …) such that each consecutive pair is connected by an edge of G. The set of all walks, denoted Walk(G) = Hom(C(ℕ), G), is equipped with the product topology ∏ₙ V(G), which captures the natural notion that two walks are close when they agree on a long initial segment.

The authors then endow the category Graph of directed graphs with a Quillen model structure. Weak equivalences are precisely those graph morphisms f : X → Y that induce a bijection on walk sets, i.e. the map Walk(X) → Walk(Y) is a set‑theoretic bijection. Cofibrations are taken to be epimorphisms (surjective on vertices and edges) and fibrations are monomorphisms (injective on vertices and edges). With these choices every object is both fibrant and cofibrant, and the model‑category axioms (2‑out‑of‑3, retract, lifting, factorisation) are verified in detail. Consequently the homotopy category Ho(Graph) is obtained by formally inverting the weak equivalences.

A central result is that Walk(G) is a complete invariant of Ho(Graph): two graphs X and Y are isomorphic in the homotopy category if and only if their walk spaces are bijectively (and hence homeomorphically, after equipping the product topology) equivalent. This shows that the combinatorial data of all infinite walks completely determines the homotopy type in this model structure.

To capture finer topological information, the authors introduce a “finite‑level” homotopy category Hoₙ(Graph). For each natural number n they consider the subspace Walkₙ(G) consisting of walks that become eventually constant after step n. A morphism is an n‑level weak equivalence if it induces a bijection on Walkₙ. As n grows, Hoₙ(Graph) approximates Ho(Graph), and the tower of categories encodes increasingly detailed information about the topology of walk spaces.

The paper’s most novel construction is the basal graph associated to any directed graph X. The basal graph B is obtained by collapsing vertices of X that cannot be distinguished by any walk: two vertices are identified if every walk that passes through one also passes through the other in the same positions. Formally, one partitions the vertex set into walk‑equivalence classes and contracts each class to a single vertex, preserving the induced edge structure. The natural projection p : X → B is a weak equivalence, and B is minimal with this property. Moreover, when B is separated (no two distinct vertices share the same set of walks), B is unique up to isomorphism. The basal graph is therefore a homotopy invariant of the model structure.

The authors compare the basal graph with the classical zeta series ζ_G(t) = exp(∑_{k≥1} |Walk_k(G)| t^k/k) of a finite graph, which encodes the numbers of periodic walks of each length. While the zeta series is a coarse invariant (different graphs can share the same zeta series), the basal graph refines it: two graphs with identical zeta series may have non‑isomorphic basal graphs, and consequently lie in distinct homotopy classes. Thus the basal graph provides a strictly finer classification tool.

Finally, the paper proves a strong equivalence theorem for finite walkable graphs—graphs that are finite and in which every vertex lies on at least one infinite walk. If the basal graph B of such a graph X is separated, then the following statements are equivalent:

1. The walk spaces Walk(X) and Walk(B) are topologically conjugate (there exists a homeomorphism intertwining the shift maps).
2. X and B are homotopy‑equivalent in the Quillen model structure (i.e., there exists a zig‑zag of weak equivalences between them).

The proof uses the projection p : X → B, the separation condition, and the fact that weak equivalences preserve the shift dynamics on walk spaces. This result bridges the categorical homotopy notion with the dynamical‑systems concept of topological conjugacy, showing that for a large class of finite graphs the two perspectives coincide.

In summary, the paper accomplishes the following:

• Defines walks as morphisms from the ℕ‑Cayley graph and equips the set of walks with a natural product topology.
• Constructs a Quillen model structure on the category of directed graphs where weak equivalences are exactly the walk‑preserving bijections.
• Determines the homotopy category Ho(Graph) and introduces a finite‑level refinement Hoₙ(Graph) that respects the topology of walk spaces.
• Introduces the basal graph, proves its existence, uniqueness (under separation), and shows it is a homotopy invariant finer than the zeta series.
• Establishes that for finite walkable graphs with separated basal graphs, topological conjugacy of walk spaces is equivalent to homotopy equivalence in the model structure.

These contributions provide a robust categorical language for symbolic dynamics on graphs, open avenues for computing homotopy invariants of networked dynamical systems, and suggest extensions to weighted, undirected, or higher‑dimensional combinatorial structures.