Generalizations of Sturmian sequences associated with N-continued fraction algorithms

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 sequencesω(x,N) and ωˆ(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 ωˆ(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.