(S,w)-Gap Shifts and Their Entropy - Cristian Ramirez; Somers, Amy (The PUMP Journal of Undergraduate Research, 2025)
The S-gap shifts have a dynamically and combinatorially rich structure. Dynamical properties of the S-gap shift can be related to the properties of the set S. There is a well known formula for the entropy of an S-gap shift. Fix a point w of the full shift on 1,2,…,k. We introduce the (S,w)-gap shift which is a generalization of the S-gap shift consisting of sequences over the alphabet 0,1,…,k in which any two 0's are separated by a word u appearing in w with length in S. We extend the formula for the entropy of the S-gap shift to a formula describing the entropy of this new class of shift spaces. Additionally, we investigate the dynamical properties including irreducibility and mixing of this generalization of the S-gap shift.