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Area-Preserving Surface Diffeomorphisms

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Abstract

We prove some generic properties for C r, r = 1,2,. . .,∞, area-preserving diffeomorphism on compact surfaces. The main result is that the union of the stable (or unstable) manifolds of hyperbolic periodic points are dense in the surface. This extends the result of Franks and Le Calvez [10] on S 2 to general surfaces. The proof uses the theory of prime ends and Lefschetz fixed point theorem.

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References

  1. Arnold,V.I.: Mathematical Methods of Classical Mechanics. Berlin-Heidelberg-New York: Springer- Verlag, 1978

  2. Birkhoff, G.D.: Dynamical Systems. Volume 9. American Math. Soc. Colloquium Publications, Providence, RI:AMS, 1966

  3. Calvez, P.Le., Yoccoz, J.C.: Un théorème d'indice pour les homéomorphismes du plan au voisinage d'un point fixe. Ann. of Math. (2) 146(2), 241–293 (1997)

    Google Scholar 

  4. Caratheodory, C.: Über die begrenzung einfach zusammenhangender gebiete. Math. Ann. 73, 323–370 (1913)

    Article  MATH  MathSciNet  Google Scholar 

  5. Conley, C., Zehnder, E.: The Birkhoff-Lewis fixed point theorem and a conjecture of V. I. Arnold. Invent. Math. 73(1), 33–49 (1983)

    Article  MathSciNet  Google Scholar 

  6. Doeff, E.: Rotation measures for homeomorphisms of the torus homotopic to a Dehn twist. Ergod. Theor. Dyn. Syst. 17, 575–591 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  7. Douady, R.: Applications du théorème des tores invariants. Thèse de troisiéme cycle, Université de Paris 7, 1992

  8. Franks, J., Rotation vectors and fixed points of area preserving surface diffeomorphisms. Trans. Amer. Math. Soc. 348(7), 2637–2662 (1996)

    Google Scholar 

  9. Franks, J.: The Conley index and non-existence of minimal homeomorphisms. Illinois J. Math. 43 457–464 (1999)

    Google Scholar 

  10. Franks, J., Le Calvez, P.: Regions of instability for non-twist maps. Ergod. Theor. Dynam. Sys. 23(1),111–141 (2003)

    Article  MathSciNet  Google Scholar 

  11. Gutierrez, C.: A counter-example to a C 2 closing lemma. Ergod. Theor. Dyn. Syst. 7(4), 509–530 (1987)

    MathSciNet  Google Scholar 

  12. Hayashi, S.: Connecting invariant manifolds and the solution of the C 1 stability and Ω-stability conjectures for flows. Ann. of Math. 145(1), 81–137 (1997)

    Article  MathSciNet  Google Scholar 

  13. Liao, S.T.: An extension of the C 1 closing lemma. Acta Sci. Natur. Univ. Pekinensis. 2, 1–41 (1979)

    Google Scholar 

  14. Mai, J.: A simpler proof of C 1 closing lemma. Scientia Sinica 10, 1021–1031 (1986)

    Google Scholar 

  15. Mather, J.: Topological proofs of some purely topological consequences of caratheodory's theory of prime ends. In: Selected Studies. Eds. Th. M. Rassias, G. M. Rassias, 1982, pp. 225–255

  16. Oliveira, F.: On the generic existence of homoclinic points. Ergodic Theory Dynamical Systems 7, 567–595 (1987)

    MATH  MathSciNet  Google Scholar 

  17. Oliveira, F.: On C ∞ genericity of homoclinic orbits. Nonlinearity 13, 653–662 (2000)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  18. Pixton, D.: Planar homoclinic points. J. Differ. Eq. 44, 1365–382 (1982)

    MathSciNet  Google Scholar 

  19. Poincaré, H.: Les méthodes nouvelles de la mécanique céleste. Paris, 1892

  20. Pugh, C.: The closing lemma. Amer. J. Math. 89, 956–1021 (1967)

    MATH  MathSciNet  Google Scholar 

  21. Pugh, C., Robinson, C.; The C 1 closing lemma, including hamiltonians. Ergod. Theor. Dyn. Sys. 3, 261–313 (1983)

  22. Robinson, C.: Generic properties of conservative systems, i, ii. Amer. J. Math. 92, 562–603, 897–906 (1970)

    MATH  MathSciNet  Google Scholar 

  23. Robinson, C.: Closing stable and unstable manifolds on the two-sphere. Proc. Amer. Math. Soc. 41, 299–303 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  24. Siegel, C.L., Moser, J.K.: Lectures on Celestial Mechanics. Berlin-Heidelberg-New York: Springer, 1971

  25. Smale, S.: Mathematical problems for the next century. Math. Intelligencer 20(2), 7–15 (1998)

    Article  MathSciNet  Google Scholar 

  26. Takens, F.: Homoclinic points in conservative systems. Invent. Math. 18, 267–292 (1972)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  27. Wen, L., Xia, Z.: A basic C 1 perturbation theorem. J. Differ. Eqs. 154(2), 267–283 (1999)

    Article  Google Scholar 

  28. Wen, L., Xia, Z.: On C 1 connecting lemmas. Trans. Amer. Math. Soc. 352,(10) (2000)

  29. Xia, Z.: Homoclinic points in symplectic and volume-preserving diffeomorphism. Commun. Math. Phys. 177, 435–449 (1996)

    Article  ADS  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Northwestern University, Evanston, Il, 60208, USA

    Zhihong Xia

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  1. Zhihong Xia
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Correspondence to Zhihong Xia.

Additional information

Communicated by G. Gallavotti

Research supported in part by National Science Foundation.

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Xia, Z. Area-Preserving Surface Diffeomorphisms. Commun. Math. Phys. 263, 723–735 (2006). https://doi.org/10.1007/s00220-005-1514-3

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