Follower, Predecessor, and Extender Set Sequences of $\beta$-Shifts

Given a one-dimensional shift $X$, let $|F_X(n)|$ be the number of follower sets of words of length $n$ in $X$, and $|P_X(n)|$ be the number of predecessor sets of words of length $n$ in $X$. We call the sequence $\{|F_X(n)|\}_{n \in \mathbb{N}}$ the follower set sequence of the shift $X$, and $\{|P_X(n)|\}_{n \in \mathbb{N}}$ the predecessor 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 examine achievable differences in limiting behavior of follower, predecessor, and extender set sequences. This is done through the classical $\beta$-shifts. We show that the follower set sequences of $\beta$-shifts must grow at most linearly in $n$, while the predecessor and extender set sequences may demonstrate exponential growth rate in $n$, depending on choice of $\beta$.