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H 1 $S$-adic sequences: a bridge between dynamics, arithmetic, and geometry

Auteurs : Thuswaldner, Jörg (Auteur de la Conférence)
CIRM (Editeur )

Résumé : Based on work done by Morse and Hedlund (1940) it was observed by Arnoux and Rauzy (1991) that the classical continued fraction algorithm provides a surprising link between arithmetic and diophantine properties of an irrational number $\alpha$, the rotation by $\alpha$ on the torus $\mathbb{T} = \mathbb{R}/\mathbb{Z}$, and combinatorial properties of the well known Sturmian sequences, a class of sequences on two letters with low subword complexity.
It has been conjectured since the early 1990ies that this correspondence carries over to generalized continued fraction algorithms, rotations on higher dimensional tori, and so-called $S$-adic sequences generated by substitutions. The idea of working towards this generalization is known as Rauzy’s program. Although, starting with Rauzy (1982) a number of examples for such a generalization was devised, Cassaigne, Ferenczi, and Zamboni (2000) came up with a counterexample that showed the limitations of such a generalization.
Nevertheless, recently Berthé, Steiner, and Thuswaldner (2016) made some further progress on Rauzy’s program and were able to set up a generalization of the above correspondences. They proved that the above conjecture is true under certain natural conditions. A prominent role in this generalization is played by tilings induced by generalizations of the classical Rauzy fractal introduced by Rauzy (1982).
Another idea which is related to the above results goes back to Artin (1924), who observed that the classical continued fraction algorithm and its natural extension can be viewed as a Poincaré section of the geodesic flow on the space $SL_2(\mathbb{Z}) \ SL_2(\mathbb{R})$. Arnoux and Fisher (2001) revisited Artin’s idea and showed that the above mentioned correspondence between continued fractions, rotations, and Sturmian sequences can be interpreted in a very nice way in terms of an extension of this geodesic flow which they called the scenery flow. Currently, Arnoux et al. are setting up elements of a generalization of this connection as well.
It is the aim of my series of lectures to review the above results.

Codes MSC :
11B83 - Special sequences and polynomials
11K50 - Metric theory of continued fractions
37B10 - Symbolic dynamics
52C23 - Quasicrystals, aperiodic tilings
53D25 - Geodesic flows

 Informations sur la Vidéo Réalisateur : Hennenfent, Guillaume Langue : Anglais Date de publication : 28/11/2017 Date de captation : 21/11/2017 Collection : Research schools Format : MP4 Durée : 01:38:52 Domaine : Number Theory ; Dynamical Systems & ODE ; Geometry Audience : Chercheurs ; Etudiants Science Cycle 2 ; Doctorants , Post - Doctorants Download : https://videos.cirm-math.fr/2017-11-21_Thuswaldner.mp4 Informations sur la rencontre Nom du congrès : Jean-Morlet chair - Research school: Tiling dynamical system / Chaire Jean-Morlet - École de recherche : Pavages et systèmes dynamiques Organisteurs Congrès : Akiyama, Shigeki ; Arnoux, PierreDates : 20/11/2017 - 24/11/2017 Année de la rencontre : 2017 URL Congrès : https://akiyama-arnoux.weebly.com/school.htmlCitation Data DOI : 10.24350/CIRM.V.19248803 Cite this video as: Thuswaldner, Jörg (2017). $S$-adic sequences: a bridge between dynamics, arithmetic, and geometry. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19248803 URI : http://dx.doi.org/10.24350/CIRM.V.19248803

### Voir aussi

Bibliographie

1. Arnoux, P., & Fisher, A.M. (2001). The scenery flow for geometric structures on the torus: The linear setting. Chinese Annals of Mathematics. Series B, 22(4), 427-470 - http://dx.doi.org/10.1142/S0252959901000425

2. Arnoux, P., & Rauzy, G. (1991). Geometric representation of sequences of complexity 2n+1. Bulletin de la Société Mathématique de France, 119(2), 199-215 - http://dx.doi.org/10.24033/bsmf.2164

3. Berthé, V., Minervino, M., Steiner, W., & Thuswaldner, J. (2016). The S-adic Pisot conjecture on two letters. Topology and its Applications, 205, 47-57 - http://dx.doi.org/10.1016/j.topol.2016.01.019

4. Cassaigne, J., Ferenczi, S., & Zamboni, L.Q. (2000). Imbalances in Arnoux-Rauzy sequences. Annales de l’Institut Fourier, 50(4), 1265-1276 - http://dx.doi.org/10.5802/aif.1792

5. Morse, M., & Hedlund, G.A. (1940). Symbolic dynamics II. Sturmian trajectories. American Journal of Mathematics, 62, 1-42 - http://dx.doi.org/10.2307/2371431

6. Rauzy, G. (1982). Nombres algébriques et substitutions. Bulletin de la Société Mathématique de France, 110, 147-178 - http://dx.doi.org/10.24033/bsmf.1957

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