Convergence and combinatorics of the Reverse algorithm

TL;DR

This paper analyzes the Reverse multidimensional continued fraction algorithm, proving its ergodicity and exponential convergence, and explores the combinatorics and balance properties of associated $S$-adic languages.

Contribution

It establishes ergodicity and exponential convergence of the Reverse algorithm and characterizes balanced languages generated by related substitutions.

Findings

Reverse algorithm is ergodic and exponentially convergent.

Almost all generated languages are balanced.

A concrete family of balanced languages is characterized by a simple substitution condition.

Abstract

We study the Reverse algorithm, a multidimensional continued fraction algorithm, which is not unimodular. We show that the Reverse algorithm is ergodic and, by proving that its second Lyapunov exponent is negative, that it is a.e. exponentially convergent. In addition to that, we attach substitutions to this algorithm and study the $S$-adic languages generated by sequences of these substitutions. The negativity of the second Lyapunov exponent implies that almost all of these languages are balanced. By a thorough study of the combinatorics of the substitutions, we are even able to obtain a concrete generic family of balanced languages that is characterized in terms of a simple condition on the underlying sequence of substitutions.