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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.
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 exchang
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
We classify simple groups that act by birational transformations on compact complex Kahler surfaces. Moreover, we show that every finitely generated simple group that acts non-trivially by birational transformations on a projective surface over an arbitrary field is finite.