Geodesic
Cutting sequences on Bouw-Möller surfaces: an 𝒮-adic characterization
[Suites de coupage sur les surfaces de Bouw-Möller: une caractérisation 𝒮-adique]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 52 (2019) no. 4, pp. 927-1023.

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We consider a symbolic coding for geodesics on the family of Veech surfaces (translation surfaces rich with affine symmetries) recently discovered by Bouw and Möller. These surfaces, as noticed by Hooper, can be realized by cutting and pasting a collection of semi-regular polygons. We characterize the set of symbolic sequences (cutting sequences) that arise by coding linear trajectories by the sequence of polygon sides crossed. We provide a full characterization for the closure of the set of cutting sequences, in the spirit of the classical characterization of Sturmian sequences and the recent characterization of Smillie-Ulcigrai of cutting sequences of linear trajectories on regular polygons. The characterization is in terms of a system of finitely many substitutions (also known as an 𝒮-adic presentation), governed by a one-dimensional continued fraction-like map. As in the Sturmian and regular polygon case, the characterization is based on renormalization and the definition of a suitable combinatorial derivation operator. One of the novelties is that derivation is done in two steps, without directly using Veech group elements, but by exploiting an affine diffeomorphism that maps a Bouw-Möller surface to the dual Bouw-Möller surface in the same Teichmüller disk. As a technical tool, we crucially exploit the presentation of Bouw-Möller surfaces via Hooper diagrams.

On considère un codage symbolique des géodésiques sur une famille de surfaces de Veech (surfaces de translation riches en symétries affines) récemment découverte par Bouw et Möller. Ces surfaces, comme l'a remarqué Hooper, peuvent être réalisées en coupant et collant une collection de polygones semi-réguliers. Dans cet article, on caractérise l'ensemble des suites symboliques (« suites de coupage » ) qui correspondent au codage de trajectoires linéaires, à l'aide de la suite des côtés des polygones croisés. On donne une caractérisation complète de l'adhérence de l'ensemble des suites de coupage, dans l'esprit de la caractérisation classique des suites sturmiennes et de la récente caractérisation par Smillie-Ulcigrai des suites de coupage des trajectoires linéaires dans les polygones réguliers. La caractérisation est donnée en termes d'un système fini de substitutions (connu aussi sous le nom de présentation 𝒮-adique), réglé par une transformation unidimensionnelle qui ressemble à l'algorithme de fraction continue. Comme dans le cas sturmien et dans celui des polygones réguliers, la caractérisation est basée sur la renormalisation et sur la définition d'un opérateur combinatoire de dérivation approprié. Une des nouveautés est que la dérivation se fait en deux étapes, sans utiliser directement les éléments du groupe de Veech, mais en utilisant un difféomorphisme affine qui envoie une surface de Bouw-Möller vers sa surface « duale », qui est dans le même disque de Teichmüller. Un outil technique utilisé est la présentation des surfaces de Bouw-Möller par les diagrammes de Hooper.

Publié le :
DOI : 10.24033/asens.2401
Classification : 37B10, 37E35; 11J70, 37D40.
Keywords: Cutting sequences, translation surfaces, Bouw-Möller surfaces, renormalization for Veech surfaces, S-adic systems, substitutions, linear complexity sequences.
Mots-clés : Suites de coupage, surfaces de translation, surfaces de Bouw-Möller, renormalisation pour une surface de Veech, systèmes S-adiques, substitutions, suites de complexité linéaire. 
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     title = {Cutting sequences  on {Bouw-M\"oller} surfaces:  an $\mathcal {S}$-adic characterization},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {927--1023},
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Davis, Diana; Pasquinelli, Irene; Ulcigrai, Corinna. Cutting sequences  on Bouw-Möller surfaces:  an $\mathcal {S}$-adic characterization. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 52 (2019) no. 4, pp. 927-1023. doi : 10.24033/asens.2401. http://geodesic.mathdoc.fr/articles/10.24033/asens.2401/

Arnoux, P.; Berthé, V., Substitutions in dynamics, arithmetics and combinatorics (Lecture Notes in Math.), Volume 1794, Springer, 2002, pp. 363-374 | DOI | MR

Arnoux, P.; Hubert, P. Fractions continues sur les surfaces de Veech, J. Anal. Math., Volume 81 (2000), pp. 35-64 (ISSN: 0021-7670) | MR | Zbl | DOI

Aulicino, D.; Nguyen, D.-M. Rank two affine submanifolds in H(2,2) and H(3,1) , Geom. Topol., Volume 20 (2016), pp. 2837-2904 (ISSN: 1465-3060) | MR | Zbl | DOI

Aulicino, D.; Nguyen, D.-M.; Wright, A. Classification of higher rank orbit closures in H odd (4) , J. Eur. Math. Soc. (JEMS), Volume 18 (2016), pp. 1855-1872 (ISSN: 1435-9855) | MR | Zbl | DOI

Arnoux, P., Substitutions in dynamics, arithmetics and combinatorics (Lecture Notes in Math.), Volume 1794, Springer, 2002, pp. 143-198 | DOI | MR

Berthé, V.; Delecroix, V., Numeration and substitution 2012 (RIMS Kôkyûroku Bessatsu, B46), Res. Inst. Math. Sci. (RIMS), Kyoto, 2014, pp. 81-123 | MR | Zbl

Bainbridge, M.; Habegger, P.; Möller, M. Teichmüller curves in genus three and just likely intersections in 𝐆mn×𝐆an , Publ. Math. Inst. Hautes Études Sci., Volume 124 (2016), pp. 1-98 (ISSN: 0073-8301) | MR | Zbl | mathdoc-id | DOI

Bouw, I. I.; Möller, M. Teichmüller curves, triangle groups, and Lyapunov exponents, Ann. of Math., Volume 172 (2010), pp. 139-185 (ISSN: 0003-486X) | MR | Zbl | DOI

Calta, K. Veech surfaces and complete periodicity in genus two, J. Amer. Math. Soc., Volume 17 (2004), pp. 871-908 (ISSN: 0894-0347) | MR | Zbl | DOI

Coven, E. M.; Hedlund, G. A. Sequences with minimal block growth, Math. Systems Theory, Volume 7 (1973), pp. 138-153 (ISSN: 0025-5661) | MR | Zbl | DOI

Christoffel, E. B. Observato arithmetica, Annali di Mathematia, Volume 6 (1875), pp. 148-152 | JFM | DOI

Davis, D. Cutting sequences, regular polygons, and the Veech group, Geom. Dedicata, Volume 162 (2013), pp. 231-261 (ISSN: 0046-5755) | MR | Zbl | DOI

Davis, D. Cutting sequences on translation surfaces, New York J. Math., Volume 20 (2014), pp. 399-429 http://nyjm.albany.edu:8000/j/2014/20_399.html (ISSN: 1076-9803) | MR | Zbl

Davis, D.; Fuchs, D.; Tabachnikov, S. Periodic trajectories in the regular pentagon, Mosc. Math. J., Volume 11 (2011), p. 439-461, 629 (ISSN: 1609-3321) | MR | Zbl

Ferenczi, S. Rank and symbolic complexity, Ergodic Theory Dynam. Systems, Volume 16 (1996), pp. 663-682 (ISSN: 0143-3857) | MR | Zbl | DOI

Gutkin, E.; Judge, C. The geometry and arithmetic of translation surfaces with applications to polygonal billiards, Math. Res. Lett., Volume 3 (1996), pp. 391-403 (ISSN: 1073-2780) | MR | Zbl | DOI

Hooper, W. P. Another Veech triangle, Proc. Amer. Math. Soc., Volume 141 (2013), pp. 857-865 (ISSN: 0002-9939) | MR | Zbl | DOI

Hooper, W. P. Grid graphs and lattice surfaces, Int. Math. Res. Not., Volume 2013 (2013), pp. 2657-2698 (ISSN: 1073-7928) | MR | Zbl | DOI

Hooper, W. P. An infinite surface with the lattice property I: Veech groups and coding geodesics, Trans. Amer. Math. Soc., Volume 366 (2014), pp. 2625-2649 (ISSN: 0002-9947) | MR | Zbl | DOI

Hooper, W. P. The invariant measures of some infinite interval exchange maps, Geom. Topol., Volume 19 (2015), pp. 1895-2038 (ISSN: 1465-3060) | MR | Zbl | DOI

Kenyon, R.; Smillie, J. Billiards on rational-angled triangles, Comment. Math. Helv., Volume 75 (2000), pp. 65-108 (ISSN: 0010-2571) | MR | Zbl | DOI

Leininger, C. J. On groups generated by two positive multi-twists: Teichmüller curves and Lehmer's number, Geom. Topol., Volume 8 (2004), pp. 1301-1359 (ISSN: 1465-3060) | MR | Zbl | DOI

Lind, D.; Marcus, B., Cambridge Univ. Press, 1995, 495 pages (ISBN: 0-521-55124-2; 0-521-55900-6) | MR | Zbl | DOI

Lanneau, E.; Nguyen, D.-M. Teichmüller curves generated by Weierstrass Prym eigenforms in genus 3 and genus 4, J. Topol., Volume 7 (2014), pp. 475-522 (ISSN: 1753-8416) | MR | Zbl | DOI

Lanneau, E.; Nguyen, D.-M.; Wright, A. Finiteness of Teichmüller curves in non-arithmetic rank 1 orbit closures, Amer. J. Math., Volume 139 (2017), pp. 1449-1463 (ISSN: 0002-9327) | MR | Zbl | DOI

Lothaire, M., Encyclopedia of Mathematics and its Applications, 90, Cambridge Univ. Press, 2002, 504 pages (ISBN: 0-521-81220-8) | MR | Zbl | DOI

McMullen, C. T. Billiards and Teichmüller curves on Hilbert modular surfaces, J. Amer. Math. Soc., Volume 16 (2003), pp. 857-885 (ISSN: 0894-0347) | MR | Zbl | DOI

McMullen, C. T. Teichmüller curves in genus two: the decagon and beyond, J. reine angew. Math., Volume 582 (2005), pp. 173-199 (ISSN: 0075-4102) | MR | Zbl | DOI

McMullen, C. T. Prym varieties and Teichmüller curves, Duke Math. J., Volume 133 (2006), pp. 569-590 (ISSN: 0012-7094) | MR | Zbl | DOI

Morse, M.; Hedlund, G. A. Symbolic dynamics II. Sturmian trajectories, Amer. J. Math., Volume 62 (1940), pp. 1-42 (ISSN: 0002-9327) | MR | JFM | DOI

Matheus, C.; Wright, A. Hodge-Teichmüller planes and finiteness results for Teichmüller curves, Duke Math. J., Volume 164 (2015), pp. 1041-1077 (ISSN: 0012-7094) | MR | Zbl | DOI

Nguyen, D.-M.; Wright, A. Non-Veech surfaces in ℋ hyp (4) are generic, Geom. Funct. Anal., Volume 24 (2014), pp. 1316-1335 (ISSN: 1016-443X) | MR | Zbl | DOI

Pasquinelli, I. Cutting sequences in Veech surfaces (preprint arXiv:1507.02469 )

Series, C. The geometry of Markoff numbers, Math. Intelligencer, Volume 7 (1985), pp. 20-29 (ISSN: 0343-6993) | MR | Zbl | DOI

Series, C. The modular surface and continued fractions, J. London Math. Soc., Volume 31 (1985), pp. 69-80 (ISSN: 0024-6107) | MR | Zbl | DOI

Series, C., Ergodic theory, symbolic dynamics, and hyperbolic spaces (Trieste, 1989) (Oxford Sci. Publ.), Oxford Univ. Press, 1991, pp. 125-151 | MR | Zbl

Smith, H. J. S. Note on continued fractions, Messenger of Mathematics, Volume 6 (1877), pp. 1-14 | JFM

Smillie, J.; Ulcigrai, C., Dynamical numbers—interplay between dynamical systems and number theory (Contemp. Math.), Volume 532, Amer. Math. Soc., 2010, pp. 29-65 | MR | Zbl | DOI

Smillie, J.; Ulcigrai, C. Beyond Sturmian sequences: coding linear trajectories in the regular octagon, Proc. Lond. Math. Soc., Volume 102 (2011), pp. 291-340 (ISSN: 0024-6115) | MR | Zbl | DOI

Thurston, W. P. On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.), Volume 19 (1988), pp. 417-431 (ISSN: 0273-0979) | MR | Zbl | DOI

Veech, W. A. Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. math., Volume 97 (1989), pp. 553-583 (ISSN: 0020-9910) | MR | Zbl | DOI

Vershik, A. M. A theorem on the Markov periodical approximation in ergodic theory, J. Sov. Math., Volume 28 (1985), pp. 667-674 | Zbl | DOI

Vorobets, Y. B. Plane structures and billiards in rational polygons: the Veech alternative, Uspekhi Mat. Nauk, Volume 51 (1996), pp. 3-42 (ISSN: 0042-1316) | MR | Zbl | DOI

Ward, C. C. Calculation of Fuchsian groups associated to billiards in a rational triangle, Ergodic Theory Dynam. Systems, Volume 18 (1998), pp. 1019-1042 (ISSN: 0143-3857) | MR | Zbl | DOI

Wright, A. Schwarz triangle mappings and Teichmüller curves: the Veech-Ward-Bouw-Möller curves, Geom. Funct. Anal., Volume 23 (2013), pp. 776-809 (ISSN: 1016-443X) | MR | Zbl | DOI

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