S-limited shifts

TL;DR

This paper investigates the construction, properties, and classification of $ ext{S}$-limited shifts, a type of symbolic dynamical system, providing conditions for their complexity, ergodicity, and conjugacy.

Contribution

It introduces the concept of $ ext{S}$-limited shifts, derives formulas for their entropy, and characterizes their ergodic and conjugacy properties.

Findings

Conditions for $ ext{S}$-limited shifts to be of finite type or sofic.

Exact entropy formulas for $ ext{S}$-limited shifts.

Intrinsic ergodicity of $ ext{S}$-limited shifts and their factors.

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.