Follower, predecessor, and extender set sequences of $ \beta $-shifts

Given a one-dimensional shift $ X $ and a word $ v $ in the language of $ X $, the follower set of $ v $ is the set of all finite words which can legally follow $ v $ in some point of $ X $. The predecessor set of $ v $ is the set of all finite words which can legally precede $ v $ in some point of $ X $. We construct the follower set sequence of $ X $ by recording, for each $ n $, the number of distinct follower sets of words of length $ n $ in $ X $. We construct the predecessor set sequence of $ X $ by recording, for each $ n $, the number of distinct predecessor sets of words of length $ n $ in $ X $. Extender sets are a generalization of follower sets (see [6]), 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, first introduced in [10]. 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 $.