Classical homogeneous multidimensional continued fraction algorithms are ergodic

TL;DR

This paper proves that a class of multidimensional continued fraction algorithms, including Rauzy induction and Selmer algorithms, are ergodic with respect to Lebesgue measure, extending understanding of their long-term statistical behavior.

Contribution

It establishes ergodicity for homogeneous multidimensional continued fraction algorithms of Rauzy induction type and Selmer algorithms, a significant advancement in their theoretical understanding.

Findings

Rauzy induction type algorithms are ergodic with respect to Lebesgue measure.

Selmer algorithms are also proven to be ergodic.

The results extend ergodic theory to a broad class of multidimensional continued fractions.

Abstract

Homogeneous continued fraction algorithms are multidimensional generalizations of the classical Euclidean algorithm, the dissipative map $$ (x_1,x_2) \in \mathbb{R}_+^2 \longmapsto \left\{\begin{array}{ll} (x_1 - x_2, x_2), & \mbox{if $x_1 \geq x_2$} (x_1, x_2 - x_1), & \mbox{otherwise.} \end{array} \right. $$ We focus on those which act piecewise linearly on finitely many copies of positive cones which we call Rauzy induction type algorithms. In particular, a variation Selmer algorithm belongs to this class. We prove that Rauzy induction type algorithms, as well as Selmer algorithms, are ergodic with respect to Lebesgue measure.