Graph Subshifts

Pablo Arrighi (1, 2, 3) , Amélia Durbec (4) , Pierre Guillon (5)

1 QuaCS - Quantum Computation Structures
2 LMF - Laboratoire Méthodes Formelles
3 IXXI - Institut Rhône-Alpin des systèmes complexes
4 LISN - Laboratoire Interdisciplinaire des Sciences du Numérique
5 I2M - Institut de Mathématiques de Marseille

We propose a definition of graph subshifts of finite type that can be seen as extending both the notions of subshifts of finite type from classical symbolic dynamics and finitely presented groups from combinatorial group theory. These are sets of graphs that are defined by forbidding finitely many local patterns. In this paper, we focus on the question whether such local conditions can enforce a specific support graph, and thus relate the model to classical symbolic dynamics. We prove that the subshifts that contain only infinite graphs are either aperiodic, or feature no residual finiteness of their period group, yielding non-trivial examples as well as two natural undecidability theorems.

Domains: Dynamical Systems [math.DS]; Discrete Mathematics [cs.DM]; Combinatorics [math.CO]

Pablo Arrighi, Amélia Durbec, Pierre Guillon. Graph Subshifts. 27th International Workshop on Cellular Automata and Discrete Complex Systems, Jul 2021, Marseille, France. ⟨hal-03238087v3⟩