Orbits of N-expansions with a finite set of digits

For N∈ N_{≥ 2} and α∈ R such that 0<α≤N-1, we define I_α: = [α, α+ 1] and Iα-:=[α,α+1) and investigate the continued fraction map Tα:Iα→Iα-, which is defined as Tα(x):=Nx-d(x), where d: I_α→ N is defined by d(x):=⌊Nx-α⌋. For N∈ N_{≥ 7}, for certain values of α, open intervals (a, b) ⊂ I_α exist...