## 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}.

Original language | English |
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Pages (from-to) | 79-119 |

Number of pages | 41 |

Journal | Monatshefte fur Mathematik |

Volume | 198 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2022 |

## Keywords

- Continued fractions
- Dynamical systems
- Gaps