Orbits of N-expansions with a finite set of digits

Abstract

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 such that for almost every x∈ I_α there is an n∈ N for which Tαn(x)∉(a,b) for all n≥ n. These gaps (a, b) are investigated using the square Υα:=Iα×Iα-, where the orbitsTαk(x),k=0,1,2,… of numbers x∈ I_α are represented as cobwebs. The squares Υ_α are the union of fundamental regions, which are related to the cylinder sets of the map T_α, according to the finitely many values of d in T_α. In this paper some clear conditions are found under which I_α is gapless. If I_α consists of at least five cylinder sets, it is always gapless. In the case of four cylinder sets there are usually no gaps, except for the rare cases that there is one, very wide gap. Gaplessness in the case of two or three cylinder sets depends on the position of the endpoints of I_α with regard to the fixed points of I_α under T_α.