اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدNote on powers in three interval exchange transformations
171
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
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.
قيم البحث
اقرأ أيضاً
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 $12$.
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
es, 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.
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.
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.
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.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
