Generalizations of Sturmian sequences associated with $N$-continued fraction algorithms

TL;DR

This paper generalizes Sturmian sequences through $N$-continued fraction algorithms, analyzing their combinatorial properties, ergodic behavior, and providing a Farey-like map with explicit invariant measures.

Contribution

It introduces $N$-continued fraction sequences, extending Sturmian sequences, and studies their combinatorial, ergodic, and dynamical properties, including explicit measures and complexity functions.

Findings

$N$-continued fraction sequences are $C$-balanced with explicit bounds.

The sequences have well-defined factor complexity functions.

A Farey-like map for $N$-continued fractions is constructed with proven ergodicity.

Abstract

Given a positive integer $N$ and $x$ irrational between zero and one, 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 $\omega(x,N)$ and $\hat{\omega}(x,N)$, 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 $\omega(x,N)$ and $\hat{\omega}(x,N)$ 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.