(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 $S$. This interplay is particularly interesting when $S$ is not syndetic such as when $S$ is the set of prime numbers or when $S=\{2^n\}$. It is a well known result that the entropy of the $S$-gap shift is given by $h(X)=\log \lambda$, where $\lambda >0$ is the unique solution to the equation ${\sum }_{n\in S}{\lambda }^{-(n+1)}=1$. Fix a point $w$ of the full shift $\{1,2,\dots ,k{\}}^{\mathbb{Z}}$. We introduce the $(S,w)$-gap shift which is a generalization of the $S$-gap shift consisting of sequences in $\{0,1,\dots ,k{\}}^{\mathbb{Z}}$ in which any two $0$’s are separated by a word $u$ appearing in $w$ such that $\mathrm{\mid }u\mathrm{\mid }\in S$. 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.