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 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.Comment: 23 pages, 2 figure