Almost everywhere balanced sequences of complexity 2n + 1

摘要

We study ternary sequences associated with a multidimensional continued fraction algorithm introduced by the first author. The algorithm is defined by two matrices and we show that it is measurably isomorphic to the shift on the set ${1,2}^{mathbb N}$ of directive sequences. For a given set $mathcal{C}$ of two substitutions, we show that there exists a $mathcal{C}$-adic sequence for every vector of letter frequencies or, equivalently, for every directive sequence. We show that their factor complexity is at most $2n+1$ and is equal to $2n+1$ if and only if the letter frequencies are rationally independent if and only if the $mathcal{C}$-adic representation is primitive. It turns out that in this case, the sequences are dendric. We also prove that $mu$-almost every $mathcal{C}$-adic sequence is balanced, where $mu$ is any shift-invariant ergodic Borel probability measure on ${1,2}^{mathbb N}$ giving a positive measure to the cylinder $[12121212]$. We also prove that the second Lyapunov exponent of the matrix cocycle associated with the measure $mu$ is negative.looseness-1.