Generalizations of Sturmian sequences associated with N-continued fraction algorithms

Given a positive integer N and x is an element of [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 omega(x, N) and 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 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. (c) 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/).