Characterizing Follower and Extender Set Sequences

TL;DR

This paper investigates the possible patterns of follower and extender set sequences in one-dimensional sofic shifts, proving they are eventually periodic and characterizing the range of such sequences.

Contribution

It establishes that follower and extender set sequences of sofic shifts are necessarily eventually periodic and characterizes the sequences that can occur.

Findings

Sequences are necessarily eventually periodic.

A wide class of eventually periodic sequences can be realized.

Differences in limsup and liminf of sequences can be arbitrarily large under certain conditions.

Abstract

Given a one-dimensional shift $X$, let $|F_X(\ell)|$ be the number of follower sets of words of length $\ell$ in $X$. We call the sequence $\{|F_X(\ell)|\}_{\ell \in \mathbb{N}}$ the follower set sequence of the shift $X$. Extender sets are a generalization of follower sets, and we define the extender set sequence similarly. In this paper, we explore which sequences may be realized as follower set sequences and extender set sequences of one-dimensional sofic shifts. We show that any follower set sequence or extender set sequence of a sofic shift must be eventually periodic. We also show that, subject to a few constraints, a wide class of eventually periodic sequences are possible. In fact, any natural number difference in the $\limsup$ and $\liminf$ of these sequences may be achieved, so long as the $\liminf$ of the sequence is sufficiently large.