Skip to main content
SpringerLink
Log in

Forcing theory for transverse trajectories of surface homeomorphisms

  • Published:
Inventiones mathematicae Aims and scope

Abstract

This paper studies homeomorphisms of surfaces isotopic to the identity by means of purely topological methods and Brouwer theory. The main development is a novel theory of orbit forcing using maximal isotopies and transverse foliations. This allows us to derive new proofs for some known results as well as some new applications, among which we note the following: we extend Franks and Handel’s classification of zero entropy maps of \(\mathbb {S}^2\) for non-wandering homeomorphisms; we show that if f is a Hamiltonian homeomorphism of the annulus, then the rotation set of f is either a singleton or it contains zero in the interior, proving a conjecture posed by Boyland; we show that there exist compact convex sets of the plane that are not the rotation set of some torus homeomorphisms, proving a first case of the Franks–Misiurewicz conjecture; we extend a bounded deviation result relative to the rotation set to the general case of torus homeomorphisms.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20

Similar content being viewed by others

Notes

  1. In all figures in the text, we will represent the plane \(\mathbb {R}^2\) as the open disk. The reason being that in many cases we are dealing with the universal covering space of an a hyperbolic surface.

  2. in the whole text “transverse” will mean “positively transverse”.

References

  1. Atkinson, G.: Recurrence of co-cycles and random walks. J. Lond. Math. Soc. (2) 13, 486–488 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  2. Addas-Zanata, S.: Uniform bounds for diffeomorphisms of the torus and a conjecture by P. Boyland. J. Lond. Math. Soc. 91(2), 537–553 (2015)

  3. Addas-Zanata, S., Tal, F.: On generic rotationless diffeomorphisms of the annulus. Proc. Am. Math. Soc. 138, 1023–1031 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Béguin, F., Crovisier, S., Le Roux, F.: Fixed point sets of isotopies on surfaces (2016). e-print arXiv:1610.00686

  5. Boyland, P.: Topological methods in surface dynamics. Topol. Appl. 58(3), 223–298 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  6. Boyland, P., de Carvalho, A., Hall, T.: New rotation sets in a family of torus homeomorphisms. Invent. Math. 204, 895–937 (2016). https://doi.org/10.1007/s00222-015-0628-2

    Article  MathSciNet  MATH  Google Scholar 

  7. Brouwer, L.E.J.: Beweis des ebenen Translationssatzes. Math. Ann. 72, 37–54 (1912)

    Article  MathSciNet  MATH  Google Scholar 

  8. Brown, M., Kister, J.-M.: Invariance of complementary domains of a fixed point set. Proc. Am. Math. Soc. 91, 503–504 (1984)

    MathSciNet  MATH  Google Scholar 

  9. Dávalos, P.: On torus homeomorphism whose rotation set is an annulus. Math. Z. 275(3-4), 1005–1045 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dávalos, P.: On annular maps of the torus and sublinear diffusion. J. Inst. Math. Jussieu (2013). e-print arXiv:1311.0046

  11. Franks, J.: Generalizations of the Poincaré–Birkhoff theorem. Ann. Math. 128, 139–151 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  12. Franks, J.: Realizing rotation vectors for torus homeomorphisms. Trans. Am. Math. Soc. 311(1), 107–115 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  13. Franks, J.: The rotation set and periodic points for torus homeomorphisms. Dyn. Syst. Chaos 1(1), 41–48 (1995)

    MathSciNet  MATH  Google Scholar 

  14. Franks, J., Handel, M.: Entropy zero area preserving diffeomorphisms of \(\mathbb{S}^{2}\). Geom. Topol. 16(4), 2187–2284 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Franks, J., Misiurewicz, M.: Rotation sets of toral flows. Proc. Am. Math. Soc. 109(1), 243–249 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ginzburg, V., Gurel, B.: Non-contractible periodic orbits in Hamiltonian dynamics on closed symplectic manifolds. Compos. Math. 152, 1777–1799 (2016)

  17. Guelman, N., Koropecki, A., Tal, F.: A characterization of annularity for area-preserving toral homeomorphisms. Math. Z. 276, 673–689 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Haefliger, A., Reeb, G.: Variétés (non séparées) à une dimension et structures feuilletées du plan. Enseign. Math. (2) 3, 107–125 (1957)

    MATH  Google Scholar 

  19. Handel, M.: There are no minimal homeomorphisms of the multipunctured plane. Ergodic Theory Dyn. Syst. 12, 75–83 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  20. Handel, M.: Surface homeomorphism with entropy zero (unpublished)

  21. Jaulent, O.: Existence d’un feuilletage positivement transverse à un homéomorphisme de surface. Annales de l’institut Fourier 64(4), 1441–1476 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kerékjártó, B.: Vorlesungen über Topologie. I Flächentopologie. Springer, Berlin (1923)

    Book  MATH  Google Scholar 

  23. Koropecki, A.: Aperiodic invariant continua for surface homeomorphisms. Math. Z. 266, 229–236 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  24. Koropecki, A., Le Calvez, P., Nassiri, M.: Prime ends rotation numbers and periodic points. Duke Math. J. 164(3), 403–472 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  25. Koropecki, A., Tal, F.: Area-preserving irrotational diffeomorphisms of the torus with sublinear diffusion. Proc. Am. Math. Soc. 142, 3483–3490 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  26. Koropecki, A., Tal, F.: Bounded and unbounded behavior for area-preserving rational pseudo-rotations. Proc. Lond. Math. Soc. 109, 785–822 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  27. Le Calvez, P.: Une version feuilletée du théorème de translation de Brouwer. Comment. Math. Helvetici 79, 229–259 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  28. Le Calvez, P.: Une version feuilletée équivariante du théorème de translation de Brouwer. Publ. Math. Inst. Hautes Études Sci. 102, 1–98 (2005)

    Article  MATH  Google Scholar 

  29. Le Roux, F.: L’ensemble de rotation autour d’un point fixe. Astérisque Soc. Math. de France 350, 1–109 (2013)

    Google Scholar 

  30. Llibre, J., MacKay, R.S.: Rotation vectors and entropy for homeomorphisms of the torus isotopic to the identity. Ergodic Theory Dyn. Syst. 11, 115–128 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  31. Mather, J.: Invariant subsets of area-preserving homeomorphisms of surfaces. Adv. Math. Suppl. Stud. 7B, 531–561 (1981)

    MathSciNet  MATH  Google Scholar 

  32. Matsumoto, S.: Prime end rotation numbers of invariant separating continua of annular homeomorphisms. Proc. Am. Math. Soc. 140, 839–845 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  33. Matsuoka, T.: Periodic Points and Braid theory. Handbook of Topological Fixed Point Theory, pp. 171–216. Springer, Dordrecht (2005)

    Book  MATH  Google Scholar 

  34. Misiurewicz, M., Zieman, K.: Rotation sets for maps of tori. J. Lond. Math. Soc. 40(3), 490–506 (1989)

    Article  MathSciNet  Google Scholar 

  35. Pixton, D.: Planar homoclinic points. J. Differ. Equ. 44(3), 365–382 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  36. Pollicott, M.: Rotation sets for homeomorphisms and homology. Trans. Am. Math. Soc. 331, 881–894 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  37. Schwartzman, S.: Asymptotic cycles. Ann. Math. 68, 270–284 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  38. Song, W.T.: Upper and lower bounds for the minimal positive entropy of pure braids. Proc. Lond. Math. Soc. 36, 224–229 (2003)

    MATH  Google Scholar 

  39. Tal, F.A.: On non-contractible periodic orbits for surface homeomorphisms. Ergodic Theory Dyn. Syst. 36(5), 1644–1655 (2016)

Download references

Author information

Authors and Affiliations

  1. Sorbonne Universités, UPMC Univ Paris 06, Institut de Mathématiques de Jussieu-Paris Rive Gauche, UMR 7586, CNRS, Univ Paris Diderot, Sorbonne Paris Cité, 75005, Paris, France

    P. Le Calvez

  2. Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, Cidade Universitária, São Paulo, SP, 05508-090, Brazil

    F. A. Tal

Authors
  1. P. Le Calvez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. F. A. Tal
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to F. A. Tal.

Additional information

F. A. Tal was partially supported by CAPES, FAPESP and CNPq-Brasil.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Le Calvez, P., Tal, F.A. Forcing theory for transverse trajectories of surface homeomorphisms. Invent. math. 212, 619–729 (2018). https://doi.org/10.1007/s00222-017-0773-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-017-0773-x

Search

Navigation