Dynamics and topology of S-gap shifts

Let and let d0=s0 and Δ(S)={dn}n where dn=sn−sn−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 Δ(S) is eventually constant and is sofic if and only if Δ(S) is eventually periodic. We also show that there is a one-to-one correspondence between the set of all S-gap shifts and 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.