Selecta Mathematica

, Volume 24, Issue 2, pp 875–934 | Cite as

Rel leaves of the Arnoux–Yoccoz surfaces

Article
  • 16 Downloads

Abstract

We analyze the rel leaves of the Arnoux–Yoccoz translation surfaces. We show that for any genus \(\mathbf {g}\geqslant 3\), the leaf is dense in the connected component of the stratum \({\mathcal {H}}(\mathbf {g}-1 ,\mathbf {g}-1)\) to which it belongs, and the one-sided imaginary-rel trajectory of the surface is divergent. For one surface on this trajectory, namely the Arnoux–Yoccoz surface itself, the horizontal foliation is invariant under a pseudo-Anosov map (and in particular is uniquely ergodic), but for all other surfaces, the horizontal foliation is completely periodic. The appendix proves a field theoretic result needed for denseness of the leaf: for any \(n \geqslant 3\), the field extension of \({\mathbb {Q}}\) obtained by adjoining a root of \(X^n-X^{n-1}-\cdots -X-1\) has no totally real subfields other than \({\mathbb {Q}}\).

Mathematics Subject Classification

37Exx 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

This work was stimulated by insightful comments of Michael Boshernitzan, who conjectured Corollary 1.6. We thank Alex Wright for directing our attention to the case \(\mathbf {g}=2\) and for his proofs of Theorems 1.9 and 5.3. Theorem 1.4, which is a crucial step in our proof of Theorem 1.8, was proved in response to our queries by Lior Bary-Soroker, Mark Shusterman, and Umberto Zannier. We thank them for agreeing to include their results in Appendix A of this paper. We thank Ivan Dynnikov, Pascal Hubert and Sasha Skripchenko for pointing out the connections to their prior work and other insightful remarks. We are also grateful to David Aulicino, Josh Bowman, Duc-Manh Nguyen and John Smillie for useful discussions. We also are grateful to the anonymous referee for useful comments which helped to improve the paper. This collaboration was supported by BSF Grant 2010428. The first author’s work is supported by NSF Grant DMS-1500965 as well as a PSC-CUNY Award (funded by The Professional Staff Congress and The City University of New York). The second author’s work was supported by ERC starter Grant DLGAPS 279893. Appendix acknowledgements We are grateful to Barak Weiss for telling us about the problem of finding the maximal totally real subfields of number fields arising in dynamics. Special thanks go to Patrick Hooper whose computer verification of Corollary A.8 for all \(n \leqslant 1000\) greatly stimulated our work. We would also like to thank Moshe Jarden for his comments on drafts of this work. The first and second appendix authors were partially supported by the Israel Science Foundation Grant No. 952/14. The third appendix author was partially supported by the ERC-Advanced Grant “Diophantine problems” (Grant Agreement No. 267273).

References

  1. 1.
    Arnoux, P.: Un exemple de semi-conjugaison entre un échange d’intervalles et une translation sur le tore. Bull. Soc. Math. France 116, 489–500 (1988)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Arnoux, P., Yoccoz, J.C.: Construction de difféomorphismes pseudo-Anosov. C. R. Acad. Sci. Paris 292(1), 75–78 (1981)MathSciNetMATHGoogle Scholar
  3. 3.
    Avila, A., Eskin, A., Möller, M.: Symplectic and isometric \(\text{SL}_2({\mathbb{R}})\)-invariant subbundles of the Hodge bundle, preprint (2014)Google Scholar
  4. 4.
    Avila, A., Hubert, P., Skripchenko, A.: Diffusion for chaotic plane sections of 3-periodic surfaces. Invent. Math. 206(1), 109–146 (2016)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bainbridge, M.: Euler characteristics of Teichmüller curves in genus two. Geom. Topol. 11, 1887–2073 (2007)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bainbridge, M., Smillie, J., Weiss, B.: Horocycle dynamics: new invariants and eigenform loci in the stratum \({\cal{H}}(1,1)\), preprint (2016). arXiv:1603.00808
  7. 7.
    Boshernitzan, M.D.: Rank two interval exchange transformations. Ergod. Theory Dyn. Syst. 8(03), 379–394 (1988)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Bowman, J.: The complete family of Arnoux–Yoccoz surfaces. Geom. Dedic. 164(1), 113–130 (2013)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Bowman, J.: Orientation-reversing involutions of the genus 3 Arnoux-Yoccoz surface and related surfaces. In: Bonk, M., Gilman, J., Masur, H., Minsky, Y., Wolf, M. (eds.) The Tradition of Ahlfors–Bers. V, vol. 510 of Contemporary Mathematics, pp. 13–23. American Mathematical Society, Providence, RI (2010)Google Scholar
  10. 10.
    Calsamiglia, G., Deroin, B., Francaviglia, S.: A transfer principle: from periods to isoperiodic foliations, arXiv:1511.07635
  11. 11.
    De Leo, R., Dynnikov, I.A.: Geometry of plane sections of the infinite regular skew polyhedron \(\{ 4, 6|4\}\). Geom. Dedic. 138, 51–67 (2009)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Delecroix, V., Patrick Hooper, W.: sage-flatsurf. https://github.com/videlec/sage-flatsurf. Accessed 12 Aug 2016
  13. 13.
    Dynnikov, I.A.: Semiclassical motion of the electron. A proof of the Novikov conjecture in general position and counterexamples. In: Solitons, Geometry, and Topology: On the CrossroadGoogle Scholar
  14. 14.
    Dynnikov, I.A.: Stability of minimal interval exchange transformations, Conference lecture, Dynamics and Geometry in Teichmüller Space, CIRM, 7-7-2015Google Scholar
  15. 15.
    Dynnikov, I.A., Skripchenko, A.: Symmetric band complexes of thin type and chaotic sections which are not quite chaotic. Trans. Mosc. Math. Soc. 76, 251–269 (2015)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Einsiedler, M., Ward, T.: Ergodic theory with a view toward number theory, Graduate texts in math. 259 (2011)Google Scholar
  17. 17.
    Eskin, A., Mirzakhani, M.: Invariant and stationary measures for the \(\text{ SL }(2,{\mathbb{R}})\) action on moduli spaceGoogle Scholar
  18. 18.
    Eskin, A., Mirzakhani, M., Mohammadi, A.: Isolation theorems for \(\text{ SL }_2({\mathbb{R}})\)-invariant submanifolds in moduli space (preprint) (2013)Google Scholar
  19. 19.
    Fathi, A., Laudenbach, F., Poénaru, V.: Thurston’s work on surfaces, Translated from the 1979 French original by D. M. Kim and D. Margalit. Mathematical Notes, 48, Princeton University Press (2012)Google Scholar
  20. 20.
    Filip, S.: Semisimplicity and rigidity of the Kontsevich-Zorich cocycle. Invent. Math. 205(3), 617–670 (2016)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Hamenstädt, U.: Ergodicity of the absolute period foliation. Isr. J. Math. (to appear)Google Scholar
  22. 22.
    Hubert, P., Lanneau, E.: An introduction to Veech surfaces. Handbook of dynamical systems 1, 501–526 (2006)Google Scholar
  23. 23.
    Hooper, W.P., Weiss, B.: The rel leaf and real-rel ray of the Arnoux-Yoccoz surface in genus 3. arXiv:1506.06773
  24. 24.
    Hubert, P., Lanneau, E., Möller, M.: The Arnoux–Yoccoz Teichmüller disc. Geom. Func. Anal. (GAFA) 18(6), 1988–2016 (2009)CrossRefMATHGoogle Scholar
  25. 25.
    Kenyon, R., Smillie, J.: Billiards in rational-angled triangles. Comment. Math. Helv. 75, 65–108 (2000)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Kontsevich, M., Zorich, A.: Connected components of the moduli spaces of Abelian differentials with prescribed singularities. Invent. Math. 153(3), 631–678 (2003)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Lang, S.: Algebra, Graduate Texts in Mathematics. Springer, New York (2002)Google Scholar
  28. 28.
    Lelièvre, S., Weiss, B.: Surfaces with no convex presentations. GAFA 25, 1902–1936 (2015)MathSciNetMATHGoogle Scholar
  29. 29.
    Lowenstein, J.H., Poggiaspalla, G., Vivaldi, F.: Interval exchange transformations over algebraic number fields: the cubic Arnoux–Yoccoz model. Dyn. Syst. 22, 73–106 (2007)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Martin, P.A.: The Galois group of \(x^n - x^{n-1} -... - 1\). J. Pure Appl. Algebra 190(1–3), 213–223 (2004)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Masur, H.: Interval exchange transformations and measured foliations. Ann. Math. 2nd Ser. 115(1), 169–200 (1982)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Masur, H., Tabachnikov, S.: Rational billiards and flat structures. In: Handbook of dynamical systems. Enc. Math. Sci. Ser. (2001)Google Scholar
  33. 33.
    McMullen, C.T.: Teichmüller geodesics of infinite complexity. Acta Math. 191, 191–223 (2003)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    McMullen, C.T.: Dynamics of \(\text{ SL }_2({\mathbb{R}})\) over moduli space in genus two. Ann. Math. 165, 397–456 (2007)MathSciNetCrossRefGoogle Scholar
  35. 35.
    McMullen, C.T.: Foliations of Hilbert modular surfaces. Am. J. Math. 129, 183–215 (2007)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    McMullen, C.T.: Navigating moduli space with complex twists. J. Eur. Math. Soc. (JEMS) 15, 1223–1243 (2013)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    McMullen, C.T.: Moduli spaces of isoperiodic forms on Riemann surfaces. Duke Math. J. 163, 2271–2323 (2014)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    McMullen, C.T.: Cascades in the dynamics of measured foliations. Ann. Sci. l’ENS 48, 1–39 (2015)MathSciNetMATHGoogle Scholar
  39. 39.
    Minsky, Y., Weiss, B.: Cohomology classes represented by measured foliations, and Mahler’s question for interval exchanges. Ann. Sci. l’ENS 47 (2014)Google Scholar
  40. 40.
    Mirzakhani, M., Wright, A.: Full rank affine invariant submanifolds, preprint (2016). arXiv:1608.02147
  41. 41.
    Möller, M.: Variations of Hodge structures of a Teichmüller curve. J. Am. Math. Soc. 19, 327–344 (2006)CrossRefMATHGoogle Scholar
  42. 42.
    Rauzy, G.: Nombres algébriques et substitutions. Bull. Soc. Math. France 110, 147–178 (1982)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Ribes, L., Zalesskii, P.: Profinite Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics 40.  https://doi.org/10.1007/978-3-642-01642-4, \(\copyright \) Springer-Verlag Berlin Heidelberg 2010
  44. 44.
    Schmoll, M.: Spaces of elliptic differentials. In: Kolyada, S., Manin, Y.I., Ward, T. (eds.) Algebraic and topological dynamics. Cont. Math., vol. 385, pp. 303–320 (2005)Google Scholar
  45. 45.
    Smillie, J., Weiss, B.: Minimal sets for flows on moduli space. Isr. J. Math. 142, 249–260 (2004)MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Thurston, W.: On the geometry and dynamics of diffeomorphisms of surfaces. Bull. AMS (new series) 19(2), 417–431 (1988)MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Veech, W.A.: Gauss measures for transformations on the space of interval exchange maps. Ann. Math. 201–242 (1982)Google Scholar
  48. 48.
    Veech, W.A.: Measures supported on the set of uniquely ergodic directions of an arbitrary holomorohic 1-form. Ergod. Theory Dyn. Syst. 19, 1093–1109 (1999)MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Weiss, B.: Dynamics on parameter spaces: submanifold and fractal subset questions. In: Burger, M., Iozzi, A. (eds.) Rigidity in Dynamics and Geometry, pp. 425–440. Springer, Berlin (2002)CrossRefGoogle Scholar
  50. 50.
    Wright, A.: The field of definition of affine invariant submanifolds of the moduli space of abelian differentials. Geom. Top. (2014)Google Scholar
  51. 51.
    Wright, A.: Translation surfaces and their orbit closures: an introduction for a broad audience. EMS Surv. Math. Sci. 2(1), 63–108 (2015)MathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Zorich, A.: Flat surfaces. In: Cartier, P., Julia, B., Moussa, P., Vanhove, P. (eds.) Frontiers in Number Theory, Physics and Geometry. Springer, Berlin (2006)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  1. 1.City College of New York and CUNY Graduate CenterNew YorkUSA
  2. 2.Tel Aviv UniversityTel AvivIsrael

Personalised recommendations