Textile Systems on Lambda-graph Systems

The notions of symbolic matrix system and λ-graph system for a subshift are generalizations of symbolic matrix and λ-graph (= finite symbolic matrix) for a sofic shift respectively ([Doc. Math. 4(1999), 285-340]). M. Nasu introduced the notion of textile system for a pair of graph homomorphisms to study automorphisms and endomorphisms of topological Markov shifts ([Mem. Amer. Math. Soc. 546,114(1995)]). In this paper, we formulate textile systems on λ-graph systems and study automorphisms on subshifts. We will prove that for a forward automorphism φ of a subshift (Λ, σ), the automorphisms φσ, k ≥ 0, n ≥ 1 can be explicitly realized as a subshift defined by certain symbolic matrix systems coming from both the strong shift equivalence representing φ and the subshift (Λ, σ). As an application of this result, if an automorphism φ of a subshift Λ is a simple automorphism, the dynamical system (Λ, φ ◦ σ) is topologically conjugate to the subshift (Λ, σ).