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A characterization of annularity for area-preserving toral homeomorphisms

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Abstract

We prove that if an area-preserving homeomorphism of the torus in the homotopy class of the identity has a rotation set which is a nondegenerate vertical segment containing the origin, then there exists an essential invariant annulus. In particular, some lift to the universal covering has uniformly bounded displacement in the horizontal direction.

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References

  1. Addas-Zanata, S., Garcia, B., Tal, F.A.: Dynamics of homeomorphisms of the torus homotopic to Dehn twists. Ergod. Theory Dyn. Syst., Publlic online in 2012. doi:10.1017/etds.2012.156

  2. Bortolatto, R.B., Tal, F.A.: Ergodicity and annular homeomorphisms of the torus. Qual. Theory Dyn. Syst., Public online in (2012). doi:10.1007/s12346-012-0095-8

  3. Cartwright, M.L., Littlewood, J.E.: Some fixed point theorems. Ann. Math. (2) 54, 1–37 (1951)

    Article  MathSciNet  Google Scholar 

  4. Dávalos, P.: On torus homeomorphisms whose rotation set is an interval. Math. Z., Public online in (2013). doi:10.1007/s00209-013-1168-3

  5. Fathi, A.: An orbit closing proof of Brouwer’s lemma on translation arcs. Enseign. Math. (2) 33(3–4), 315–322 (1987)

    MATH  MathSciNet  Google Scholar 

  6. Franks, J.: The rotation set and periodic points for torus homeomorphisms. In: Dynamical Systems and Chaos, vol. 1 (Hachioji, 1994), World Sci. Publ., River Edge, pp. 41–48 (1995)

  7. Handel, M.: The rotation set of a homeomorphism of the annulus is closed. Commun. Math. Phys. 127(2), 339–349 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  8. Jäger, T.: The concept of bounded mean motion for toral homeomorphisms. Dyn. Syst. 24(3), 277–297 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  9. Kocsard, A., Koropecki, A.: Free curves and periodic points for torus homeomorphisms. Ergod. Theory Dyn. Syst. 28(6), 1895–1915 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  10. Kocsard, A., Koropecki, A.: A mixing-like property and inexistence of invariant foliations for minimal diffeomorphisms of the 2-torus. Proc. Am. Math. Soc. 137(10), 3379–3386 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  11. Koropecki, A., Le Calvez, P., Nassiri, M.: Prime ends rotation numbers and periodic points, eprint ArXiv:1206.4707 (2012)

  12. Koropecki, A., Tal, F.A.: Area-preserving irrotational diffeomorphisms of the torus with sublinear diffusion. Proc. Am. Math. Soc. (2013) (to appear)

  13. Koropecki, A., Tal, F.A.: Bounded and unbounded behavior for area-preserving rational pseudo-rotations. e-print ArXiv:1207.5573 (2012)

  14. Koropecki, A., Tal, F.A.: Strictly Toral dynamics. Invent. Math., Publ. online in 2013. doi:10.1007/s00222-013-0470-3

  15. Koropecki, A.: Aperiodic invariant continua for surface homeomorphisms. Math. Z. 266(1), 229–236 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  16. Mather, J.N.: Variational construction of orbits of twist diffeomorphisms. J. Am. Math. Soc. 4(2), 207–263 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  17. Misiurewicz, M., Ziemian, K.: Rotation sets for maps of tori. J. Lond. Math. Soc. 40(2), 490–506 (1989)

    Article  MathSciNet  Google Scholar 

  18. Slijepčević, S.: The shear-rotation interval of twist maps. Ergod. Theory Dyn. Syst. 22(1), 303–313 (2002)

    MATH  Google Scholar 

  19. Tal, F.A.: Transitivity and rotation sets with nonempty interior for homeomorphisms of the 2-torus. Proc. Am. Math. Soc. 140(10), 3567–3579 (2012)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. IMERL, Facultad de Ingeniería, Universidad de la República, C.C. 30, Montevideo, Uruguay

    Nancy Guelman

  2. Instituto de Matemática e Estatística, Universidade Federal Fluminense, Rua Mário Santos Braga S/N, Niteroi, RJ, 24020-140, Brasil

    Andres Koropecki

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

    Fabio Armando Tal

Authors
  1. Nancy Guelman
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  2. Andres Koropecki
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  3. Fabio Armando Tal
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Corresponding author

Correspondence to Andres Koropecki.

Additional information

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

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Guelman, N., Koropecki, A. & Tal, F.A. A characterization of annularity for area-preserving toral homeomorphisms. Math. Z. 276, 673–689 (2014). https://doi.org/10.1007/s00209-013-1218-x

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  • DOI: https://doi.org/10.1007/s00209-013-1218-x

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