Dynamics and topology of S-gap shifts

Let S = { s i ∈ N ∪ { 0 } : 0 ⩽ s i < s i + 1 , i ∈ N ∪ { 0 } } and let d 0 = s 0 and Δ ( 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 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 { r ∈ R : r ⩾ 0 } \ { 1 n : n ∈ 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.