Arnoux-Rauzy interval exchange transformations

TL;DR

This paper investigates the dynamical properties of Arnoux-Rauzy interval exchange transformations, establishing measure-theoretic isomorphisms under certain conditions and exploring their ergodic and mixing behaviors.

Contribution

It introduces a new nine-interval exchange system and proves measure-theoretic isomorphism results for Arnoux-Rauzy systems under Diophantine conditions, advancing understanding of their ergodic properties.

Findings

Measure-theoretic isomorphism holds under Diophantine conditions for almost all systems.

Interval exchanges can be non-uniquely ergodic under certain conditions.

Conditions for weak mixing in these systems are provided.

Abstract

The Arnoux-Rauzy systems are defined in \cite{ar}, both as symbolic systems on three letters and exchanges of six intervals on the circle. In connection with a conjecture of S.P. Novikov, we investigate the dynamical properties of the interval exchanges, and precise their relation with the symbolic systems, which was known only to be a semi-conjugacy; in order to do this, we define a new system which is an exchange of nine intervals on the line (it was described in \cite{abb} for a particular case). Our main result is that the semi-conjugacy determines a measure-theoretic isomorphism (between the three systems) under a diophantine (sufficient) condition, which is satisfied by almost all Arnoux-Rauzy systems for a suitable measure; but, under another condition, the interval exchanges are not uniquely ergodic and the isomorphism does not hold for all invariant measures; finally, we give conditions for these interval exchanges to be weakly mixing.