Rosen fractions and Veech groups, an overly brief introduction

Thomas A. Schmidt[1]

  • [1] Oregon State University Corvallis, OR 97331

Actes des rencontres du CIRM (2009)

  • Volume: 1, Issue: 1, page 61-67
  • ISSN: 2105-0597

Abstract

top
We give a very brief, but gentle, sketch of an introduction both to the Rosen continued fractions and to a geometric setting to which they are related, given in terms of Veech groups. We have kept the informal approach of the talk at the Numerations conference, aimed at an audience assumed to have heard of neither of the topics of the title.The Rosen continued fractions are a family of continued fraction algorithms, each gives expansions of real numbers in terms of elements of a corresponding algebraic number field. A Veech group is comprised of the Jacobians of locally affine self-maps on a “flat” surface to itself. The Rosen fractions are directly related to a certain family of (projective) matrix groups; these groups are directly related to W.  Veech’s original examples of surfaces with “optimal” dynamics.

How to cite

top

Schmidt, Thomas A.. "Rosen fractions and Veech groups, an overly brief introduction." Actes des rencontres du CIRM 1.1 (2009): 61-67. <http://eudml.org/doc/10021>.

@article{Schmidt2009,
abstract = {We give a very brief, but gentle, sketch of an introduction both to the Rosen continued fractions and to a geometric setting to which they are related, given in terms of Veech groups. We have kept the informal approach of the talk at the Numerations conference, aimed at an audience assumed to have heard of neither of the topics of the title.The Rosen continued fractions are a family of continued fraction algorithms, each gives expansions of real numbers in terms of elements of a corresponding algebraic number field. A Veech group is comprised of the Jacobians of locally affine self-maps on a “flat” surface to itself. The Rosen fractions are directly related to a certain family of (projective) matrix groups; these groups are directly related to W.  Veech’s original examples of surfaces with “optimal” dynamics.},
affiliation = {Oregon State University Corvallis, OR 97331},
author = {Schmidt, Thomas A.},
journal = {Actes des rencontres du CIRM},
language = {eng},
month = {3},
number = {1},
pages = {61-67},
publisher = {CIRM},
title = {Rosen fractions and Veech groups, an overly brief introduction},
url = {http://eudml.org/doc/10021},
volume = {1},
year = {2009},
}

TY - JOUR
AU - Schmidt, Thomas A.
TI - Rosen fractions and Veech groups, an overly brief introduction
JO - Actes des rencontres du CIRM
DA - 2009/3//
PB - CIRM
VL - 1
IS - 1
SP - 61
EP - 67
AB - We give a very brief, but gentle, sketch of an introduction both to the Rosen continued fractions and to a geometric setting to which they are related, given in terms of Veech groups. We have kept the informal approach of the talk at the Numerations conference, aimed at an audience assumed to have heard of neither of the topics of the title.The Rosen continued fractions are a family of continued fraction algorithms, each gives expansions of real numbers in terms of elements of a corresponding algebraic number field. A Veech group is comprised of the Jacobians of locally affine self-maps on a “flat” surface to itself. The Rosen fractions are directly related to a certain family of (projective) matrix groups; these groups are directly related to W.  Veech’s original examples of surfaces with “optimal” dynamics.
LA - eng
UR - http://eudml.org/doc/10021
ER -

References

top
  1. P. Arnoux, P. Hubert, Fractions continues sur les surfaces de Veech, J. Anal. Math. 81 (2000), 35–64. Zbl1029.11035MR1785277
  2. E. Artin, Ein mechanisches System mit quasi-ergodischen Bahnen, Abh. Math. Sem. Hamburg 3 (1924) 170-175 (and Collected Papers, Springer-Verlag, New York, 1982, 499-505). Zbl50.0677.11
  3. F. Berg, Dreiecksgruppen mit Spitzen in quadratischen Zahlkörpern, Abh. Math. Sem. Univ. Hamburg 55 (1985), 191–200. Zbl0581.10010MR831527
  4. E. Bogomolny and C. Schmit, Multiplicities of periodic orbit lengths for non-arithmetic models, J. Phys. A: Math. Gen. 37, (2004) 4501-4526. Zbl1050.37008MR2065941
  5. R. Bowen, C. Series, Markov maps associated with Fuchsian groups, Inst. Hautes Études Sci. Publ. Math. No. 50 (1979), 153–170. Zbl0439.30033MR556585
  6. D. Bump, Automorphic forms and representations, Cambridge Studies in Advanced Mathematics, 55. Cambridge University Press, Cambridge, 1997. Zbl0868.11022MR1431508
  7. R. Burton, C. Kraaikamp, and T.A. Schmidt, Natural extensions for the Rosen fractions, TAMS 352 (1999), 1277–1298. Zbl0938.11036MR1650073
  8. K. Calta, Veech surfaces and complete periodicity in genus two, J. Amer. Math. Soc. 17 (2004), no. 4, 871–908 Zbl1073.37032MR2083470
  9. K. Dajani, C. Kraaikamp, W. Steiner, Metrical theory for -Rosen fractions, preprint. ArXiv : 0702516. Zbl1184.28016
  10. E. Gutkin, Billiards in polygons: Survey of recent results, J. Stat. Phys. 83 (1996), 7 – 26. Zbl1081.37525MR1382759
  11. E. Gutkin and C. Judge, Affine mappings of translation surfaces: Geometry and arithmetic, Duke Math. J. 103 (2000), 191 – 213. Zbl0965.30019MR1760625
  12. E. Hanson, A. Merberg, C. Towse, and E. Yudovina, Generalized continued fractions and orbits under the action of Hecke triangle groups Acta Arith. 134 (2008), no. 4, 337–348. Zbl1219.11105MR2449157
  13. P. Hubert, T. A. Schmidt, An introduction to Veech surfaces; in: Handbook of dynamical systems. Vol. 1B, 501–526, Elsevier B. V., Amsterdam, 2006. Zbl1130.37367MR2186246
  14. I. Ivrissimtzis, D. Singerman, Regular maps and principal congruence subgroups of Hecke groups, European J. Combin. 26 (2005), no. 3-4, 437–456. Zbl1075.20022MR2116181
  15. S. Kerckhoff, H. Masur, and J. Smillie, Ergodicity of billiard flows and quadratic differentials, Ann. Math. 124 (1986), 293 – 311. Zbl0637.58010MR855297
  16. M.-L. Lang, C.-H. Lim, S.-P. Tan, Principal congruence subgroups of the Hecke groups, J. Number Theory 85 (2) (2000) 220–230. Zbl0964.11023MR1802713
  17. A. Leutbecher Über die Heckeschen Gruppen , Abh. Math. Sem. Hamb. 31 (1967), 199-205. Zbl0161.27601MR228438
  18. H. Masur and S. Tabachnikov, Rational billiards and flat structures, in Handbook on Dynamical Systems, Elsevier 2001, in press. Zbl1057.37034MR1928530
  19. D. Mayer and F. Strömberg,Symbolic dynamics for the geodesic flow on Hecke surfaces, J. Mod. Dyn. 2 (2008), no. 4, 581–627. Zbl1161.37031MR2449139
  20. C. McMullen, Teichmüller geodesics of infinite complexity, Acta Math. 191 (2003), no. 2, 191–223 Zbl1131.37052MR2051398
  21. H. Nakada, Continued fractions, geodesic flows and Ford circles, in Algorithms, Fractals and Dynamics edited by Y. Takahashi, 179–191, Plenum, 1995. Zbl0868.30005MR1402490
  22. D. Rosen, A Class of Continued Fractions Associated with Certain Properly Discontinuous Groups, Duke Math. J. 21 (1954), 549-563. Zbl0056.30703MR65632
  23. T. A. Schmidt, M. Sheingorn, Length spectra of the Hecke triangle groups, Math. Z. 220 (1995), no. 3, 369–397 Zbl0840.11019MR1362251
  24. J. Smillie, Dynamics of billiard flow in rational polygons; In: Ya. G. Sinai (ed) Dynamical Systems. Encycl. Math. Sci. Vol. 100. Math. Physics 1. Springer Verlag, 360 – 382 (2000) Zbl1323.37002
  25. J. Smillie and C. Ulcigrai Symbolic coding for linear trajectories in the regular octagon, preprint arXiv:0905.0871. 
  26. W. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. A.M.S. 19 (1988), 417 – 431. Zbl0674.57008MR956596
  27. D. Rosen, C. Towse, Continued fraction representations of units associated with certain Hecke groups, Arch. Math. (Basel) 77 (2001), no. 4, 294–302. Zbl0992.11034MR1853543
  28. M. Viana, Ergodic theory of interval exchange maps Rev. Mat. Complut. 19 (2006), no. 1, 7–100. Zbl1112.37003MR2219821
  29. W.A. Veech, Teichmuller curves in modular space, Eisenstein series, and an application to triangular billiards, Inv. Math. 97 (1989), 553 – 583. Zbl0676.32006MR1005006
  30. Ya. B. Vorobets , Plane structures and billiards in rational polyhedra: the Veech alternative (Russian), Uspekhi Mat. Nauk 51, 3–42, (1996); translation in Russian Math. Surveys 51:5 (1996), 779–817 Zbl0897.58029MR1436653
  31. L. Washington, Introduction to cyclotomic fields, GTM 83. Springer-Verlag, New York, 1997. Zbl0966.11047MR1421575
  32. A. Zorich, Flat surfaces; in: Frontiers in number theory, physics, and geometry. I, 437–583, Springer, Berlin, 2006. Zbl1129.32012MR2261104

NotesEmbed ?

top

You must be logged in to post comments.