Periodic orbits and chain-transitive sets of C1-diffeomorphisms

Sylvain Crovisier

Publications Mathématiques de l'IHÉS (2006)

  • Volume: 104, page 87-141
  • ISSN: 0073-8301

Abstract

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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).

How to cite

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Crovisier, Sylvain. "Periodic orbits and chain-transitive sets of C1-diffeomorphisms." Publications Mathématiques de l'IHÉS 104 (2006): 87-141. <http://eudml.org/doc/104221>.

@article{Crovisier2006,
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).},
author = {Crovisier, Sylvain},
journal = {Publications Mathématiques de l'IHÉS},
keywords = {-generic diffeomorphisms; periodic orbits; homoclinic classes; chain-recurrence classes; shadowing},
language = {eng},
pages = {87-141},
publisher = {Springer},
title = {Periodic orbits and chain-transitive sets of C1-diffeomorphisms},
url = {http://eudml.org/doc/104221},
volume = {104},
year = {2006},
}

TY - JOUR
AU - Crovisier, Sylvain
TI - Periodic orbits and chain-transitive sets of C1-diffeomorphisms
JO - Publications Mathématiques de l'IHÉS
PY - 2006
PB - Springer
VL - 104
SP - 87
EP - 141
AB - 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).
LA - eng
KW - -generic diffeomorphisms; periodic orbits; homoclinic classes; chain-recurrence classes; shadowing
UR - http://eudml.org/doc/104221
ER -

References

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  1. 1. F. Abdenur, C. Bonatti, S. Crovisier, Global dominated splittings and the C1 Newhouse phenomenon, Proc. Amer. Math. Soc., 134 (2006), 2229-2237 Zbl1088.37013MR2213695
  2. 2. F. Abdenur, C. Bonatti, S. Crovisier, L. Díaz, Generic diffeomorphisms on compact surfaces, Fundam. Math., 187 (2005), 127-159 Zbl1089.37032MR2214876
  3. 3. F. Abdenur and L. Díaz, Pseudo-orbit shadowing in the C1-topology, to appear in Discrete Cont. Dyn. Syst. Zbl1132.37014MR2257429
  4. 4. R. Abraham, S. Smale, Nongenericity of Ω-stability, Global analysis I, Proc. Symp. Pure Math. AMS, 14 (1970), 5-8 Zbl0215.25102
  5. 5. M.-C. Arnaud, Création de connexions en topologie C1, Ergodic Theory Dyn. Syst., 21 (2001), 339-381 Zbl0997.37007MR1827109
  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 Zbl1024.37011
  7. 7. M.-C. Arnaud, C. Bonatti, S. Crovisier, Dynamiques symplectiques génériques, Ergodic Theory Dyn. Syst., 25 (2005), 1401-1436 Zbl1084.37017MR2173426
  8. 8. C. Bonatti, S. Crovisier, Récurrence et généricité, Invent. Math., 158 (2004), 33-104 Zbl1071.37015MR2090361
  9. 9. C. Bonatti, L. Díaz, Persistent nonhyperbolic transitive diffeomorphisms, Ann. Math., 143 (1996), 357-396 Zbl0852.58066MR1381990
  10. 10. C. Bonatti, L. Díaz, On maximal transitive sets of generic diffeomorphisms, Publ. Math., Inst. Hautes Étud. Sci., 96 (2003), 171-197 Zbl1032.37011MR1985032
  11. 11. C. Bonatti, L. Díaz, E. Pujals, A C1-generic dichotomy for diffeomorphisms: weak forms of hyperbolicicity or infinitely many sinks or sources, Ann. Math., 158 (2003), 355-418 Zbl1049.37011MR2018925
  12. 12. C. Bonatti, L. Díaz, G. Turcat, Pas de “shadowing lemma” pour des dynamiques partiellement hyperboliques, C. R. Acad. Sci. Paris, 330 (2000), 587-592 Zbl0973.37016
  13. 13. R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Springer, Berlin – New York (1975) Zbl0308.28010MR442989
  14. 14. C. Conley, Isolated invariant sets and Morse index, AMS, Providence (1978) Zbl0397.34056MR511133
  15. 15. C. Carballo, C. Morales, M.-J. Pacífico, Homoclinic classes for C1-generic vector fields, Ergodic Theory Dyn. Syst., 23 (2003), 1-13 Zbl1047.37009MR1972228
  16. 16. R. Corless, S. Pilyugin, Approximate and real trajectories for generic dynamical systems, J. Math. Anal. Appl., 189 (1995), 409-423 Zbl0821.58036MR1312053
  17. 17. W. Melo, Structural stability of diffeomorphisms on two-manifolds, Invent. Math., 21 (1973), 233-246 Zbl0291.58011MR339277
  18. 18. G. Gan, L. Wen, Heteroclinic cycles and homoclinic closures for generic diffeomorphisms, J. Dyn. Differ. Equations, 15 (2003), 451-471 Zbl1034.37013MR2046726
  19. 19. S. Gonchenko, L. Shilńikov, D. Turaev, Dynamical phenomena in systems with structurally unstable Poincaré homoclinic orbits, Chaos, 6 (1996), 15-31 Zbl1055.37578MR1376892
  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 Zbl0871.58067
  21. 21. I. Kupka, Contribution à la théorie des champs génériques, Contrib. Differ. Equ., 2 (1963), 457-484 Zbl0149.41002MR165536
  22. 22. R. Mañé, Contributions to the stability conjecture, Topology, 17 (1978), 383-396 Zbl0405.58035MR516217
  23. 23. R. Mañé, An ergodic closing lemma, Ann. Math., 116 (1982), 503-540 Zbl0511.58029MR678479
  24. 24. R. Mañé, A proof of the C1 stability conjecture, Publ. Math., Inst. Hautes Étud. Sci., 66 (1988), 161-210 Zbl0678.58022MR932138
  25. 25. M. Mazur, Tolerance stability conjecture revisited, Topology Appl., 131 (2003), 33-38 Zbl1024.37012MR1982812
  26. 26. S. Newhouse, Hyperbolic limit sets, Trans. Amer. Math. Soc., 167 (1972), 125-150 Zbl0239.58009MR295388
  27. 27. S. Newhouse, Diffeomorphisms with infinitely many sinks, Topology, 13 (1974), 9-18 Zbl0275.58016MR339291
  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 Zbl0445.58022MR556584
  29. 29. K. Odani, Generic homeomorphisms have the pseudo-orbit tracing property, Proc. Amer. Math. Soc., 110 (1990), 281-284 Zbl0713.58025MR1009998
  30. 30. J. Palis, On the C1 Ω-stability conjecture, Publ. Math., Inst. Hautes Étud. Sci., 66 (1988), 211-215 Zbl0648.58019
  31. 31. J. Palis, S. Smale, Structural stability theorem, Proc. Amer. Math. Soc. Symp. Pure Math., 14 (1970), 223-232 Zbl0214.50702MR267603
  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. Zbl0790.58014MR1237641
  33. 33. J. Palis, M. Viana, High dimension diffeomorphisms displaying infinitely many periodic attractors, Ann. Math., 140 (1994), 207-250 Zbl0817.58004MR1289496
  34. 34. S. Pilyugin, Shadowing in dynamical systems, Lect. Notes Math., vol. 1706, Springer, Berlin, 1999. Zbl0954.37014MR1727170
  35. 35. C. Pugh, The closing lemma, Amer. J. Math., 89 (1967), 956-1009 Zbl0167.21803MR226669
  36. 36. C. Pugh, An improved closing lemma and a general density theorem, Amer. J. Math., 89 (1967), 1010-1021 Zbl0167.21804MR226670
  37. 37. C. Pugh, C. Robinson, The C1-closing lemma, including hamiltonians, Ergodic Theory Dyn. Syst., 3 (1983), 261-314 Zbl0548.58012MR742228
  38. 38. J. Robbin, A structural stability theorem, Ann. Math., 94 (1971), 447-493 Zbl0224.58005MR287580
  39. 39. C. Robinson, Generic properties of conservative systems, Amer. J. Math., 92 (1970), 562-603 Zbl0212.56502MR273640
  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. Zbl0275.58014MR334282
  41. 41. C. Robinson, Structural stability of C1-diffeomorphisms, J. Differ. Equ., 22 (1976), 28-73 Zbl0343.58009MR474411
  42. 42. C. Robinson, Stability theorems and hyperbolicity in dynamical systems, Rocky Mt. J. Math., 7 (1977), 425-437 Zbl0375.58016MR494300
  43. 43. N. Romero, Persistence of homoclinic tangencies in higher dimensions, Ergodic Theory Dyn. Syst., 15 (1995), 735-757 Zbl0833.58020MR1346398
  44. 44. K. Sakai, Diffeomorphisms with weak shadowing, Fundam. Math., 168 (2001), 57-75 Zbl0968.37009MR1835482
  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. Zbl0289.58013MR331431
  46. 46. M. Shub, Topologically transitive diffeomorphisms of T4, Lect. Notes Math., vol. 206, pp. 39–40, Springer, Berlin–New York, 1971. 
  47. 47. C. Simon, A 3-dimensional Abraham-Smale example, Proc. Amer. Math. Soc., 34 (1972), 629-630 Zbl0259.58006MR295391
  48. 48. S. Smale, Stable manifolds for differential equations and diffeomorphisms, Ann. Sc. Norm. Super. Pisa, 17 (1963), 97-116 Zbl0113.29702MR165537
  49. 49. S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817 Zbl0202.55202MR228014
  50. 50. F. Takens, On Zeeman’s tolerance stability conjecture, Lect. Notes Math., vol. 197, 209–219, Springer, Berlin, 1971. Zbl0217.48303
  51. 51. F. Takens, Tolerance stability, Lect. Notes Math., vol. 468, 293–304, Springer, Berlin, 1975. Zbl0321.54022MR650298
  52. 52. L. Wen, A uniform C1 connecting lemma, Discrete Contin. Dyn. Syst., 8 (2002), 257-265 Zbl1136.37317MR1877839
  53. 53. L. Wen, Z. Xia, C1 connecting lemmas, Trans. Amer. Math. Soc., 352 (2000), 5213-5230 Zbl0947.37018MR1694382
  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. Zbl0282.54024MR341534
  55. 55. G. Yau, J. Yorke, An open set of maps for which every point is absolutely non-shadowable, Proc. Amer. Math. Soc., 128 (2000), 909-918 Zbl0996.37025MR1623005

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