Textile systems on lambda-graph systems

Abstract

The notions of symbolic matrix system and $\lambda$-graph system for a subshift are generalizations of symbolic matrix and $\lambda$-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 $\lambda$-graph systems and study automorphisms on subshifts. We will prove that for a forward automorphism $\phi$ of a subshift $(\Lambda,\sigma)$, the automorphisms $\phi^k \sigma^n, k\ge 0, n\ge 1$ can be explicitly realized as a subshift defined by certain symbolic matrix systems coming from both the strong shift equivalence representing $\phi$ and the subshift $(\Lambda,\sigma)$. As an application of this result, if an automorphism $\phi$ of a subshift $\Lambda$ is a simple automorphism, the dynamical system $(\Lambda, \phi \circ \sigma)$ is topologically conjugate to the subshift $(\Lambda, \sigma).$