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 ${\{1,2\}}^{\mathbb{N}}$ 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.

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