Periodic orbits and chain-transitive sets of C1-diffeomorphisms

Article

Abstract

We prove that the chain-transitive sets of C1-generic diffeomorphisms are approximated in the Hausdorff topology by periodic orbits. This implies that the homoclinic classes are dense among the chain-recurrence classes.

This result is a consequence of a global connecting lemma, which allows to build by a C1-perturbation an orbit connecting several prescribed points. One deduces a weak shadowing property satisfied by C1-generic diffeomorphisms: any pseudo-orbit is approximated in the Hausdorff topology by a finite segment of a genuine orbit. As a consequence, we obtain a criterion for proving the tolerance stability conjecture in Diff1(M).

References

  1. 1.
    F. Abdenur, C. Bonatti and S. Crovisier, Global dominated splittings and the C1 Newhouse phenomenon, Proc. Amer. Math. Soc., 134 (2006), 2229–2237.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    F. Abdenur, C. Bonatti, S. Crovisier and L. Díaz, Generic diffeomorphisms on compact surfaces, Fundam. Math., 187 (2005), 127–159.MATHGoogle Scholar
  3. 3.
    F. Abdenur and L. Díaz, Pseudo-orbit shadowing in the C1-topology, to appear in Discrete Cont. Dyn. Syst. Google Scholar
  4. 4.
    R. Abraham and S. Smale, Nongenericity of Ω-stability, Global analysis I, Proc. Symp. Pure Math. AMS, 14 (1970), 5–8.MathSciNetGoogle Scholar
  5. 5.
    M.-C. Arnaud, Création de connexions en topologie C1, Ergodic Theory Dyn. Syst., 21 (2001), 339–381.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    M.-C. Arnaud, Approximation des ensembles ω-limites des difféomorphismes par des orbites périodiques, Ann. Sci. Éc. Norm. Supér., IV. Sér., 36 (2003), 173–190.MATHMathSciNetGoogle Scholar
  7. 7.
    M.-C. Arnaud, C. Bonatti and S. Crovisier, Dynamiques symplectiques génériques, Ergodic Theory Dyn. Syst., 25 (2005), 1401–1436.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    C. Bonatti and S. Crovisier, Récurrence et généricité, Invent. Math., 158 (2004), 33–104.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    C. Bonatti and L. Díaz, Persistent nonhyperbolic transitive diffeomorphisms, Ann. Math., 143 (1996), 357–396.MATHCrossRefGoogle Scholar
  10. 10.
    C. Bonatti and L. Díaz, On maximal transitive sets of generic diffeomorphisms, Publ. Math., Inst. Hautes Étud. Sci., 96 (2003), 171–197.Google Scholar
  11. 11.
    C. Bonatti, L. Díaz and E. Pujals, A C1-generic dichotomy for diffeomorphisms: weak forms of hyperbolicicity or infinitely many sinks or sources, Ann. Math., 158 (2003), 355–418.MATHCrossRefGoogle Scholar
  12. 12.
    C. Bonatti, L. Díaz and G. Turcat, Pas de “shadowing lemma” pour des dynamiques partiellement hyperboliques, C. R. Acad. Sci. Paris, 330 (2000), 587–592.MATHGoogle Scholar
  13. 13.
    R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lect. Notes Math., vol. 470, Springer, Berlin – New York, 1975.MATHGoogle Scholar
  14. 14.
    C. Conley, Isolated invariant sets and Morse index, CBMS Regional Conference Series in Mathematics, 38, AMS, Providence, 1978.MATHGoogle Scholar
  15. 15.
    C. Carballo, C. Morales and M.-J. Pacífico, Homoclinic classes for C1-generic vector fields, Ergodic Theory Dyn. Syst., 23 (2003), 1–13.CrossRefGoogle Scholar
  16. 16.
    R. Corless and S. Pilyugin, Approximate and real trajectories for generic dynamical systems, J. Math. Anal. Appl., 189 (1995), 409–423.MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    W. De Melo, Structural stability of diffeomorphisms on two-manifolds, Invent. Math., 21 (1973), 233–246.MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    G. Gan and L. Wen, Heteroclinic cycles and homoclinic closures for generic diffeomorphisms, J. Dyn. Differ. Equations, 15 (2003), 451–471.MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    S. Gonchenko, L. Shilńikov and D. Turaev, Dynamical phenomena in systems with structurally unstable Poincaré homoclinic orbits, Chaos, 6 (1996), 15–31.MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    S. Hayashi, Connecting invariant manifolds and the solution of the C1-stability and Ω-stability conjectures for flows, Ann. Math., 145 (1997), 81–137 and 150 (1999), 353–356.MATHCrossRefGoogle Scholar
  21. 21.
    I. Kupka, Contribution à la théorie des champs génériques, Contrib. Differ. Equ., 2 (1963), 457–484 and 3 (1964), 411–420.MathSciNetGoogle Scholar
  22. 22.
    R. Mañé, Contributions to the stability conjecture, Topology, 17 (1978), 383–396.MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    R. Mañé, An ergodic closing lemma, Ann. Math., 116 (1982), 503–540.CrossRefGoogle Scholar
  24. 24.
    R. Mañé, A proof of the C1 stability conjecture, Publ. Math., Inst. Hautes Étud. Sci., 66 (1988), 161–210.MATHGoogle Scholar
  25. 25.
    M. Mazur, Tolerance stability conjecture revisited, Topology Appl., 131 (2003), 33–38.MATHMathSciNetGoogle Scholar
  26. 26.
    S. Newhouse, Hyperbolic limit sets, Trans. Amer. Math. Soc., 167 (1972), 125–150.MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    S. Newhouse, Diffeomorphisms with infinitely many sinks, Topology, 13 (1974), 9–18.MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    S. Newhouse, The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms, Publ. Math., Inst. Hautes Étud. Sci., 50 (1979), 101–151.MATHMathSciNetGoogle Scholar
  29. 29.
    K. Odani, Generic homeomorphisms have the pseudo-orbit tracing property, Proc. Amer. Math. Soc., 110 (1990), 281–284.MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    J. Palis, On the C1 Ω-stability conjecture, Publ. Math., Inst. Hautes Étud. Sci., 66 (1988), 211–215.MATHMathSciNetGoogle Scholar
  31. 31.
    J. Palis and S. Smale, Structural stability theorem, Proc. Amer. Math. Soc. Symp. Pure Math., 14 (1970), 223–232.MathSciNetGoogle Scholar
  32. 32.
    J. Palis and F. Takens, Hyperbolicity & sensitive chaotic dynamics at homoclinic bifurcations, Cambridge Studies in Advanced Mathematics, 35, Cambridge University Press, Cambridge, 1993.Google Scholar
  33. 33.
    J. Palis and M. Viana, High dimension diffeomorphisms displaying infinitely many periodic attractors, Ann. Math., 140 (1994), 207–250.MATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    S. Pilyugin, Shadowing in dynamical systems, Lect. Notes Math., vol. 1706, Springer, Berlin, 1999.Google Scholar
  35. 35.
    C. Pugh, The closing lemma, Amer. J. Math., 89 (1967), 956–1009.MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    C. Pugh, An improved closing lemma and a general density theorem, Amer. J. Math., 89 (1967), 1010–1021.MATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    C. Pugh, and C. Robinson, The C1-closing lemma, including hamiltonians, Ergodic Theory Dyn. Syst., 3 (1983), 261–314.MATHMathSciNetGoogle Scholar
  38. 38.
    J. Robbin, A structural stability theorem, Ann. Math., 94 (1971), 447–493.CrossRefMathSciNetGoogle Scholar
  39. 39.
    C. Robinson, Generic properties of conservative systems, Amer. J. Math., 92 (1970), 562–603 and 897–906.MATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    C. Robinson, Cr-structural stability implies Kupka–Smale, Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971), pp. 443–449, Academic Press, New York, 1973.Google Scholar
  41. 41.
    C. Robinson, Structural stability of C1-diffeomorphisms, J. Differ. Equ., 22 (1976), 28–73.MATHCrossRefGoogle Scholar
  42. 42.
    C. Robinson, Stability theorems and hyperbolicity in dynamical systems, Rocky Mt. J. Math., 7 (1977), 425–437.MATHCrossRefGoogle Scholar
  43. 43.
    N. Romero, Persistence of homoclinic tangencies in higher dimensions, Ergodic Theory Dyn. Syst., 15 (1995), 735–757.MATHMathSciNetGoogle Scholar
  44. 44.
    K. Sakai, Diffeomorphisms with weak shadowing, Fundam. Math., 168 (2001), 57–75.MATHGoogle Scholar
  45. 45.
    M. Shub, Stability and genericity for diffeomorphisms, Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971), pp. 493–514, Academic Press, New York, 1973.Google Scholar
  46. 46.
    M. Shub, Topologically transitive diffeomorphisms of T4, Lect. Notes Math., vol. 206, pp. 39–40, Springer, Berlin–New York, 1971.Google Scholar
  47. 47.
    C. Simon, A 3-dimensional Abraham-Smale example, Proc. Amer. Math. Soc., 34 (1972), 629–630.MATHCrossRefMathSciNetGoogle Scholar
  48. 48.
    S. Smale, Stable manifolds for differential equations and diffeomorphisms, Ann. Sc. Norm. Super. Pisa, 17 (1963), 97–116.MATHMathSciNetGoogle Scholar
  49. 49.
    S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747–817.CrossRefMathSciNetGoogle Scholar
  50. 50.
    F. Takens, On Zeeman’s tolerance stability conjecture, Lect. Notes Math., vol. 197, 209–219, Springer, Berlin, 1971.Google Scholar
  51. 51.
    F. Takens, Tolerance stability, Lect. Notes Math., vol. 468, 293–304, Springer, Berlin, 1975.Google Scholar
  52. 52.
    L. Wen, A uniform C1 connecting lemma, Discrete Contin. Dyn. Syst., 8 (2002), 257–265.MATHMathSciNetCrossRefGoogle Scholar
  53. 53.
    L. Wen and Z. Xia, C1 connecting lemmas, Trans. Amer. Math. Soc., 352 (2000), 5213–5230.MATHCrossRefMathSciNetGoogle Scholar
  54. 54.
    W. White, On the tolerance stability conjecture, Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971), pp. 663–665, Academic Press, New York, 1973.Google Scholar
  55. 55.
    G. Yau and J. Yorke, An open set of maps for which every point is absolutely non-shadowable, Proc. Amer. Math. Soc., 128 (2000), 909–918.CrossRefMathSciNetGoogle Scholar

Copyright information

© IHES and Springer-Verlag 2006

Authors and Affiliations

  1. 1.CNRS – Laboratoire Analyse, Géométrie et Applications, UMR 7539, Institut GaliléeUniversité Paris 13VilletaneuseFrance

Personalised recommendations