S-limited shifts

Abstract

In this paper, we explore the construction and dynamical properties of $\mathcal{S}$-limited shifts. An $S$-limited shift is a subshift defined on a finite alphabet $\mathcal{A}=\{1,\dots ,p\}$ by a set $\mathcal{S}=\{S_1,\dots ,S_p\}$, where $S_i\subseteq \mathbb{N}$ describes the allowable lengths of blocks in which the corresponding letter may appear. We give conditions for which an $\mathcal{S}$-limited shift is a subshift of finite type or sofic. We give an exact formula for finding the entropy of such a shift and show that an $\mathcal{S}$-limited shift and its factors must be intrinsically ergodic. Finally, we give some conditions for which two such shifts can be conjugate, and additional information about conjugate $\mathcal{S}$-limited shifts.