Almost everywhere balanced sequences of complexity 2n + 1

Abstract

. 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 } N of directive sequences. For a given set C of two substitutions, we show that there exists a C -adic sequence for every vector of letter frequencies or, equivalently, for every directive sequence. We show that their factor complexity is at most 2 n +1 and is 2 n +1 if and only if the letter frequencies are rationally independent if and only if the C -adic representation is primitive. It turns out that in this case, the sequences are dendric. We also prove that µ -almost every C -adic sequence is balanced, where µ is any shift-invariant ergodic Borel probability measure on { 1 , 2 } 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 µ is negative.