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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.
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In this paper, we study distortion in the group $mathcal A$ of Affine Interval Exchange Transformations (AIET). We prove that any distorted element $f$ of $mathcal A$, has an iterate $f^ k$ that is conjugate by an element of $mathcal A$ to a product of infinite order restricted rotations, with pairwise disjoint supports. As consequences we prove that no Baumslag-Solitar group, $BS(m,n)$ with $vert m vert eq vert n vert$, acts faithfully by elements of $mathcal A$, every finitely generated nilpotent group of $mathcal A$ is virtually abelian and there is no distortion element in $mathcal A_{mathbb Q}$, the subgroup of $mathcal A$ consisting of rational AIETs.
There exist uniquely ergodic affine interval exchange transformations of [0,1] with flips having wandering intervals and such that the support of the invariant measure is a Cantor set.
We study the existence of transitive exchange maps with flips defined on the unit circle. We provide a complete answer to the question of whether there exists a transitive exchange map of the unit circle defined on n subintervals and having f flips.
We study repetitions in infinite words coding exchange of three intervals with permutation (3,2,1), called 3iet words. The language of such words is determined by two parameters $varepsilon,ell$. We show that finiteness of the index of 3iet words is equivalent to boundedness of the coefficients of the continued fraction of $varepsilon$. In this case we also give an upper and lower estimate on the index of the corresponding 3iet word.
We describe a generalization of a result of Boshernitzan and Carroll: an extension of Lagranges 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 transforma- tion whose lengths are defined over a quadratic field and applying the two-sided Rauzy induction, one can obtain only a finite number of new transformations up to homothety.
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