Dynamics and Topology of S-gap Shifts

Abstract: Let $S={s_i\in\mathbb N\cup{0}:0\leq s_i<s_{i+1}}$ and let $d_{0}=s_{0}$ and $\Delta(S)={d_{n}}{n}$ where $d{n}=s_{n}-s_{n-1}$. In this note, we show that an $S$-gap shift is subshift of finite type (SFT) if and only if $S$ is finite or cofinite, is almost-finite-type (AFT) if and only if $\Delta(S)$ is eventually constant and is sofic if and only if $\Delta(S)$ is eventually periodic. We also show that there is a one-to-one correspondence between the set of all $S$-gap shifts and ${r \in \mathbb R: r \geq 0}\backslash {\frac{1}{n}: n \in {\mathbb N}}$ up to conjugacy. This enables us to induce a topology and measure structure on the set of all $S$-gaps. By using this, we give the frequency of certain $S$-gap shifts with respect to their dynamical properties.