Dynamique des échanges d’intervalles des groupes de Higman-Thompson V_{r,m}

Abstract

Dynamique des échanges d'intervalles des groupes de Higman-ThompsonL'accès aux articles de la revue « Annales de l'institut Fourier » (http://aif.cedram.org/), implique l'accord avec les conditions générales d'utilisation (http://aif.cedram.org/legal/). Toute reproduction en tout ou partie de cet article sous quelque forme que ce soit pour tout usage autre que l'utilisation à fin strictement personnelle du copiste est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Abstract. -In this paper, we study the dynamics of affine interval exchange transformations, whose slopes are integer powers of the same integer m, and whose cuts and their images are rational. We prove that such a map has very simple dynamics: all its orbits are proper and it has at least one periodic orbit or periodic cycle. As a corollary, we obtain that a distortion element of the Higman-Thompson group Vr,m is of finite order.

Distortion in groups of affine interval exchange transformations

“…Last two theorems were proved by Higman for Thompson's group V (see [14] and [25], chapter 2). This theorem extends to all groups of rational AIET, results of Burillo-Cleary-Röver (see [6]) and Hmili-Liousse (see [15]) on non existence of distortion in Thompson's groups V n . The main consequence of this theorem is that any group G containing distortion elements has no faithful actions as rational affine interval exchange transformations.…”

Non-smoothability for a class of groups of piecewise linear homeomorphisms of the interval

“…-Actually, Hyde and Tatch Moore were only considering maps T which induce a circle homeomorphism (after identifying the endpoints of I), but it can be interesting, although we have not yet found significant applications, to look at more general locally defined transformations T , such as generalized interval exchange maps of I, or doubling maps of the circle obtained by identifying the endpoints of I. For instance, results from [3,12] allow to describe the dynamics of affine interval exchange transformations locally defined by elements of Thompson's group F , and for certain generalized interval exchange maps some Herman-Yoccoz theory has been developed by Ghazouani and Ulcigrai [8].…”

Distortion and Tits alternative in smooth mapping class groups

“…The previous construction is useful for our purposes because it is known there are no distorted elements in Thompson's group V n (see Bleak-Collin-et al [2], Sec.8, see also Hmili-Liousse [20], Corollary 1.10). By Proposition 4.2, this implies there are no distorted elements in V 2 .…”