Exactness of the Euclidean algorithm and of the Rauzy induction on the space of interval exchange transfomations

TL;DR

This paper proves that the Euclidean algorithm and Rauzy induction, both key tools in number theory and dynamical systems, are exact transformations, enhancing understanding of their ergodic properties.

Contribution

It establishes the exactness of the Euclidean algorithm and Rauzy induction, providing new insights into their ergodic behavior in the context of interval exchange transformations.

Findings

Both maps are dissipative and ergodic with respect to Lebesgue measure.

The maps are proven to be exact transformations.

Results deepen understanding of their ergodic properties.

Abstract

The two-dimensional homogeneous Euclidean algorithm is the central motivation for the definition of the classical multidimensional continued fraction algorithms, as Jacobi-Perron, Poincar\'e, Brun and Selmer algorithms. The Rauzy induction, a generalization of the Euclidean algorithm, is a key tool in the study of interval exchange transformations. Both maps are known to be dissipative and ergodic with respect to Lebesgue measure. Here we prove that they are exact.