Textile systems on lambda-graph systems

Abstract

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 $ϕ^k σ^n, k\ge 0, n\ge 1$ 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 $(Λ, ϕ\circ σ)$ is topologically conjugate to the subshift $(Λ, σ).$