2022/09/02 by Niels Langeveld, Langeveld, Niels, Lucía Rossi +3
Computer Science · Mathematics · #11J70 #37A44 #68R15 #Combinatorics (math.CO) #Computability, Logic, AI Algorithms #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2209.00978
openalex publication_date 2022/09/02 · openalex created_date 2022/09/06 · openalex updated_date 2026/07/28
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.