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 \(\mathcal{A} = \{1, \ldots,p\}\) by a set \(\mathcal{S} = \{S_1, \ldots, S_p\}\), 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.