ARNOUX-RAUZY INTERVAL EXCHANGE TRANSFORMATIONS

Pierre Arnoux (1) , Julien Cassaigne (1) , Sébastien Ferenczi (1) , Pascal Hubert (1)

1 I2M - Institut de Mathématiques de Marseille

The Arnoux-Rauzy systems are defined in [5], 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 (it was described in [3] for a particular case). Our main result is that the semi-conjugacy determines a measure-theoretic isomorphism under an explicit (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.

Domaines: Mathématiques [math]

Pierre Arnoux, Julien Cassaigne, Sébastien Ferenczi, Pascal Hubert. ARNOUX-RAUZY INTERVAL EXCHANGE TRANSFORMATIONS. Annali della Scuola Normale Superiore di Pisa, In press. ⟨hal-02120138⟩