On the second Lyapunov exponent of some multidimensional continued fraction algorithms

TL;DR

This paper investigates the convergence properties of multidimensional continued fraction algorithms, proving the negativity of the second Lyapunov exponent for Selmer's algorithm in two dimensions and suggesting most algorithms lose strong convergence in higher dimensions.

Contribution

It provides the first proof of the negativity of the second Lyapunov exponent for Selmer's algorithm and offers heuristic insights into the convergence behavior of various multidimensional algorithms.

Findings

Second Lyapunov exponent of Selmer's algorithm is negative.

Most classical algorithms lose strong convergence in high dimensions.

Arnoux-Rauzy algorithm is an exception, but only on measure zero sets.

Abstract

We study the strong convergence of certain multidimensional continued fraction algorithms. In particular, in the two-dimensional case, we prove that the second Lyapunov exponent of Selmer's algorithm is negative and bound it away from zero. Moreover, we give heuristic results on several other continued fraction algorithms. Our results indicate that all classical multidimensional continued fraction algorithms cease to be strongly convergent for high dimensions. The only exception seems to be the Arnoux-Rauzy algorithm which, however, is defined only on a set of measure zero.