Almost everywhere balanced sequences of complexity $2n+1$

TL;DR

This paper investigates ternary sequences generated by a multidimensional continued fraction algorithm, establishing their complexity, balance properties, and ergodic characteristics, with implications for combinatorics and dynamical systems.

Contribution

It introduces a new connection between a multidimensional continued fraction algorithm and balanced sequences with complexity $2n+1$, including their measure-theoretic and dynamical properties.

Findings

Sequences have complexity exactly $2n+1$ if and only if they are primitive and have rationally independent frequencies.

Almost every $ ext{C}$-adic sequence is balanced under a shift-invariant ergodic measure.

The second Lyapunov exponent of the associated matrix cocycle is negative.

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 $2n+1$ 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.