Dynamique des échanges d’intervalles des groupes de Higman-Thompson V r,m  [ Dynamics of affine interval transformations of Higman-Thompson groups V r,m  ]
Annales de l'Institut Fourier, Volume 64 (2014) no. 4, p. 1477-1491
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 V r,m is of finite order.
Dans cet article, nous étudions la dynamique des échanges d’intervalles affines dont les pentes sont des puissances d’un même entier m et dont les coupures et leurs images sont des rationnels. Nous montrons qu’une telle application a une dynamique très simple  : toutes ses orbites sont propres et elle possède au moins une orbite périodique ou un cycle périodique. Comme corollaire de ce résultat, nous montrons que les éléments de distortion dans les groupes de Higman-Thompson V r,m sont ceux d’ordre fini.
Received : 2012-09-10
Accepted : 2012-12-17
DOI : https://doi.org/10.5802/aif.2887
Classification:  37E05,  37C85,  20F05
Keywords: Affine interval exchange transformations, groups, periodic orbits, distortion elements
@article{AIF_2014__64_4_1477_0,
     author = {Hmili, Hadda and Liousse, Isabelle},
     title = {Dynamique des \'echanges d'intervalles  des groupes de Higman-Thompson $V\_{r,m}$},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {64},
     number = {4},
     year = {2014},
     pages = {1477-1491},
     doi = {10.5802/aif.2887},
     zbl = {06387314},
     mrnumber = {3329670},
     language = {fr},
     url = {https://aif.centre-mersenne.org/item/AIF_2014__64_4_1477_0}
}
Hmili, Hadda; Liousse, Isabelle. Dynamique des échanges d’intervalles  des groupes de Higman-Thompson $V_{r,m}$. Annales de l'Institut Fourier, Volume 64 (2014) no. 4, pp. 1477-1491. doi : 10.5802/aif.2887. https://aif.centre-mersenne.org/item/AIF_2014__64_4_1477_0/

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