(S,w)-Gap Shifts and Their Entropy

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.