Suffix Conjugates for a Class of Morphic Subshifts

Abstract

Let A be a finite alphabet and f : A * → A * a morphism with an iterative fixed point f ω (α), where α ∈ A. Consider the subshift \(({\mathcal X}, T)\), where \({\mathcal X}\) is the shift orbit closure of f ω (α) and \(T\colon{\mathcal X} \rightarrow{\mathcal X}\) is the shift operation. Let S be a finite alphabet that is in bijective correspondence via a mapping c with the set of nonempty suffixes of the images f(a) for a ∈ A. Let \({\mathcal S}\subset S^{\mathbb N}\) be the set of infinite words s = (s n ) n ≥ 0 such that \(\pi({\rm s}) := c(s_{0})f({c(s_{1})})f^{2}({c(s_{2})})\cdots \in{\mathcal X}\). We show that if f is primitive and f(A) is a suffix code, then there exists a mapping \(H : {\mathcal S} \rightarrow{\mathcal S}\) such that \(({\mathcal S}, H)\) is a topological dynamical system and \(\pi :({\mathcal S}, H) \rightarrow ({\mathcal X}, T)\) is a conjugacy. We call \(({\mathcal S}, H)\) the suffix conjugate of (χ, T). Furthermore, in the special case when f is the Fibonacci or the Thue-Morse morphism, we show that \(({\mathcal S}, T)\) is a sofic shift, that is, the language of \({\mathcal S}\) is regular.