Suffix conjugates for a class of morphic subshifts

Abstract

Let $A$ be a finite alphabet and $f:~A^{\ast }\rightarrow A^{\ast }$ be a morphism with an iterative fixed point $f^{{\it\omega}}({\it\alpha})$, where ${\it\alpha}\in A$. Consider the subshift $({\mathcal{X}},T)$, where ${\mathcal{X}}$ is the shift orbit closure of $f^{{\it\omega}}({\it\alpha})$ and $T:~{\mathcal{X}}\rightarrow {\mathcal{X}}$ is the shift map. Let $S$ be a finite alphabet that is in bijective correspondence via a mapping $c$ with the set of non-empty suffixes of the images $f(a)$ for $a\in A$. Let ${\mathcal{S}}\subset S^{\mathbb{N}}$ be the set of infinite words $\mathbf{s}=(s_{n})_{n\geq 0}$ such that ${\it\pi}(\mathbf{s}):=c(s_{0})f(c(s_{1}))f^{2}(c(s_{2}))\cdots \in {\mathcal{X}}$. We show that if $f$ is primitive, $f^{{\it\omega}}({\it\alpha})$ is aperiodic, 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 ${\it\pi}:~({\mathcal{S}},H)\rightarrow ({\mathcal{X}},T)$ is a conjugacy; we call $({\mathcal{S}},H)$ the suffix conjugate of $({\mathcal{X}},T)$. In the special case where $f$ is the Fibonacci or Thue–Morse morphism, we show that the subshift $({\mathcal{S}},T)$ is sofic, that is, the language of ${\mathcal{S}}$ is regular.