## Abstract

Given a positive integer N and x ∈ [0, 1] \ Q, an N-continued

fraction expansion of x is defined analogously to the classical

continued fraction expansion, but with the numerators being

all equal to N. Inspired by Sturmian sequences, we introduce

the N-continued fraction sequences ω(x, N) and ω(x, N),

which are related to the N-continued fraction expansion of

x. They are infinite words over a two letter alphabet obtained

as the limit of a directive sequence of certain substitutions,

hence they are S-adic sequences. When N = 1, we are in the

case of the classical continued fraction algorithm, and obtain

the well-known Sturmian sequences. We show that ω(x, N)

and ω(x, N) are C-balanced for some explicit values of C

and compute their factor complexity function. We also obtain

uniform word frequencies and deduce unique ergodicity of the

associated subshifts. Finally, we provide a Farey-like map for

N-continued fraction expansions, which provides an additive

version of N-continued fractions, for which we prove ergodicity

and give the invariant measure explicitly

fraction expansion of x is defined analogously to the classical

continued fraction expansion, but with the numerators being

all equal to N. Inspired by Sturmian sequences, we introduce

the N-continued fraction sequences ω(x, N) and ω(x, N),

which are related to the N-continued fraction expansion of

x. They are infinite words over a two letter alphabet obtained

as the limit of a directive sequence of certain substitutions,

hence they are S-adic sequences. When N = 1, we are in the

case of the classical continued fraction algorithm, and obtain

the well-known Sturmian sequences. We show that ω(x, N)

and ω(x, N) are C-balanced for some explicit values of C

and compute their factor complexity function. We also obtain

uniform word frequencies and deduce unique ergodicity of the

associated subshifts. Finally, we provide a Farey-like map for

N-continued fraction expansions, which provides an additive

version of N-continued fractions, for which we prove ergodicity

and give the invariant measure explicitly

Originalsprache | Englisch |
---|---|

Seiten (von - bis) | 49-83 |

Seitenumfang | 35 |

Fachzeitschrift | Journal of number theory |

Jahrgang | 250.2023 |

Ausgabenummer | September |

Frühes Online-Datum | 20 Apr. 2023 |

DOIs | |

Publikationsstatus | Veröffentlicht - Sept. 2023 |