2021/07/14 by Jaap de Jonge, de Jonge, J., Cor Kraaikamp +3
Computer Science · Mathematics · #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2107.06722
openalex publication_date 2021/07/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a natural number N≥ 2 and a real α such that 0 < α≤ √(N)-1, we define Iα:=[α,α+1] and Iα-:=[α,α+1) and investigate the continued fraction map Tα:Iα→ Iα-, which is defined as Tα(x):= N/x-d(x), where d(x):= \lfloor N/x -α \rfloor. For all natural N ≥ 7, for certain values of α, open intervals (a,b) ⊂ Iα exist such that for almost every x ∈ Iα there is an natural number n0 for which Tαn(x) ∉ (a,b) for all n≥ n0. These gaps (a,b) are investigated in the square Υα:=Iα× Iα-, where the orbits Tαk(x), k=0,1,2,… of numbers x ∈ Iα are represented as cobwebs. The squares Υα are the union of fundamental regions, which are related to the cylinder sets of the map Tα, according to the finitely many values of d in Tα. In this paper some clear conditions are found under which Iα is gapless. When Iα 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 Iα with regard to the fixed points of Iα under Tα.