Interval exchanges, admissibility and branching Rauzy induction

Decoding Rauzy Induction: Bufetov’s Question

Answering a question posed by Alexander Bufetov, it is proved that if a pair of (i.d.o.c.) interval exchanges on [0, 1) have identical sequences of visitation matrices with respect to Rauzy induction, then the exchanges are homeomorphically conjugate and, in particular, are governed by the same permutation. 2000 Math. Subj. Class. 37E05, 37B10.

Dynamical generalizations of the Lagrange spectrum

We compute two invariants of topological conjugacy, the upper and lower limits of the inverse of Boshernitzan's ne n , where e n is the smallest measure of a cylinder of length n, for three families of symbolic systems, the natural codings of rotations and three-interval exchanges and the Arnoux-Rauzy systems. The sets of values of these invariants for a given family of systems generalize the L...

Decay of Correlations for the Rauzy–Veech–Zorich Induction Map on the Space of Exchanges of Four Intervals

Regular Interval Exchange Transformations over a Quadratic Field

We describe a generalization of a result of Boshernitzan and Carroll: an extension of Lagrange’s Theorem on continued fraction expansion of quadratic irrationals to interval exchange transformations. In order to do this, we use a two-sided version of the Rauzy induction. In particular, we show that starting from an interval exchange transformation whose lengths are defined over a quadratic fiel...

Exponential decay of correlations for the Rauzy-Veech-Zorich induction map

We prove exponential mixing for the Rauzy-Veech-Zorich induction map on the space of interval exchange transformations (Theorem 3).