ﻻ يوجد ملخص باللغة العربية
Let $mathcal G$ be the group of all Interval Exchange Transformations. Results of Arnoux-Fathi ([Arn81b]), Sah ([Sah81]) and Vorobets ([Vor17]) state that $mathcal G_0$ the subgroup of $mathcal G$ generated by its commutators is simple. In [Arn81b], Arnoux proved that the group $overline{mathcal G}$ of all Interval Exchange Transformations with flips is simple. We establish that every element of $overline{mathcal G}$ has a commutator length not exceeding $6$. Moreover, we give conditions on $mathcal G$ that guarantee that the commutator lengths of the elements of $mathcal G_0$ are uniformly bounded, and in this case for any $gin mathcal G_0$ this length is at most $5$. As analogous arguments work for the involution length in $overline{mathcal G}$, we add an appendix whose purpose is to prove that every element of $overline{mathcal G}$ has an involution length not exceeding $12$.
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
Let $I=[0,1)$ and $mathcal{PC}(I)$ [resp. $mathcal{PC}^+(I)$] be the quotient group of the group of all piecewise continuous [resp. piecewise continuous and orientation preserving] bijections of $I$ by its normal subgroup consisting in elements with
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