Periodic orbits and chain-transitive sets of C 1 -diffeomorphisms
Publications Mathématiques de l'IHÉS, Tome 104 (2006), pp. 87-141.

We prove that the chain-transitive sets of C 1 -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 C 1 -perturbation an orbit connecting several prescribed points. One deduces a weak shadowing property satisfied by C 1 -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 Diff 1 (M).

@article{PMIHES_2006__104__87_0,
     author = {Crovisier, Sylvain},
     title = {Periodic orbits and chain-transitive sets of $C^1$-diffeomorphisms},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {87--141},
     publisher = {Springer},
     volume = {104},
     year = {2006},
     doi = {10.1007/s10240-006-0002-4},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1007/s10240-006-0002-4/}
}
TY  - JOUR
AU  - Crovisier, Sylvain
TI  - Periodic orbits and chain-transitive sets of $C^1$-diffeomorphisms
JO  - Publications Mathématiques de l'IHÉS
PY  - 2006
SP  - 87
EP  - 141
VL  - 104
PB  - Springer
UR  - http://archive.numdam.org/articles/10.1007/s10240-006-0002-4/
DO  - 10.1007/s10240-006-0002-4
LA  - en
ID  - PMIHES_2006__104__87_0
ER  - 
%0 Journal Article
%A Crovisier, Sylvain
%T Periodic orbits and chain-transitive sets of $C^1$-diffeomorphisms
%J Publications Mathématiques de l'IHÉS
%D 2006
%P 87-141
%V 104
%I Springer
%U http://archive.numdam.org/articles/10.1007/s10240-006-0002-4/
%R 10.1007/s10240-006-0002-4
%G en
%F PMIHES_2006__104__87_0
Crovisier, Sylvain. Periodic orbits and chain-transitive sets of $C^1$-diffeomorphisms. Publications Mathématiques de l'IHÉS, Tome 104 (2006), pp. 87-141. doi : 10.1007/s10240-006-0002-4. http://archive.numdam.org/articles/10.1007/s10240-006-0002-4/

1. F. Abdenur, C. Bonatti, S. Crovisier, Global dominated splittings and the C1 Newhouse phenomenon, Proc. Amer. Math. Soc., 134 (2006), 2229-2237 | MR | Zbl

2. F. Abdenur, C. Bonatti, S. Crovisier, L. Díaz, Generic diffeomorphisms on compact surfaces, Fundam. Math., 187 (2005), 127-159 | MR | Zbl

3. F. Abdenur and L. Díaz, Pseudo-orbit shadowing in the C1-topology, to appear in Discrete Cont. Dyn. Syst. | MR | Zbl

4. R. Abraham, S. Smale, Nongenericity of Ω-stability, Global analysis I, Proc. Symp. Pure Math. AMS, 14 (1970), 5-8 | Zbl

5. M.-C. Arnaud, Création de connexions en topologie C1 , Ergodic Theory Dyn. Syst., 21 (2001), 339-381 | MR | Zbl

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 | Numdam | Zbl

7. M.-C. Arnaud, C. Bonatti, S. Crovisier, Dynamiques symplectiques génériques, Ergodic Theory Dyn. Syst., 25 (2005), 1401-1436 | MR | Zbl

8. C. Bonatti, S. Crovisier, Récurrence et généricité, Invent. Math., 158 (2004), 33-104 | MR | Zbl

9. C. Bonatti, L. Díaz, Persistent nonhyperbolic transitive diffeomorphisms, Ann. Math., 143 (1996), 357-396 | MR | Zbl

10. C. Bonatti, L. Díaz, On maximal transitive sets of generic diffeomorphisms, Publ. Math., Inst. Hautes Étud. Sci., 96 (2003), 171-197 | EuDML | Numdam | MR | Zbl

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 | MR | Zbl

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 | Zbl

13. R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Springer, Berlin - New York (1975) | MR | Zbl

14. C. Conley, Isolated invariant sets and Morse index, AMS, Providence (1978) | MR | Zbl

15. C. Carballo, C. Morales, M.-J. Pacífico, Homoclinic classes for C1-generic vector fields, Ergodic Theory Dyn. Syst., 23 (2003), 1-13 | MR | Zbl

16. R. Corless, S. Pilyugin, Approximate and real trajectories for generic dynamical systems, J. Math. Anal. Appl., 189 (1995), 409-423 | MR | Zbl

17. W. Melo, Structural stability of diffeomorphisms on two-manifolds, Invent. Math., 21 (1973), 233-246 | EuDML | MR | Zbl

18. G. Gan, L. Wen, Heteroclinic cycles and homoclinic closures for generic diffeomorphisms, J. Dyn. Differ. Equations, 15 (2003), 451-471 | MR | Zbl

19. S. Gonchenko, L. Shilńikov, D. Turaev, Dynamical phenomena in systems with structurally unstable Poincaré homoclinic orbits, Chaos, 6 (1996), 15-31 | MR | Zbl

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 | Zbl

21. I. Kupka, Contribution à la théorie des champs génériques, Contrib. Differ. Equ., 2 (1963), 457-484 | MR | Zbl

22. R. Mañé, Contributions to the stability conjecture, Topology, 17 (1978), 383-396 | MR | Zbl

23. R. Mañé, An ergodic closing lemma, Ann. Math., 116 (1982), 503-540 | MR | Zbl

24. R. Mañé, A proof of the C1 stability conjecture, Publ. Math., Inst. Hautes Étud. Sci., 66 (1988), 161-210 | EuDML | Numdam | MR | Zbl

25. M. Mazur, Tolerance stability conjecture revisited, Topology Appl., 131 (2003), 33-38 | MR | Zbl

26. S. Newhouse, Hyperbolic limit sets, Trans. Amer. Math. Soc., 167 (1972), 125-150 | MR | Zbl

27. S. Newhouse, Diffeomorphisms with infinitely many sinks, Topology, 13 (1974), 9-18 | MR | Zbl

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 | EuDML | Numdam | MR | Zbl

29. K. Odani, Generic homeomorphisms have the pseudo-orbit tracing property, Proc. Amer. Math. Soc., 110 (1990), 281-284 | MR | Zbl

30. J. Palis, On the C1 Ω-stability conjecture, Publ. Math., Inst. Hautes Étud. Sci., 66 (1988), 211-215 | EuDML | Numdam | Zbl

31. J. Palis, S. Smale, Structural stability theorem, Proc. Amer. Math. Soc. Symp. Pure Math., 14 (1970), 223-232 | MR | Zbl

32. J. Palis and F. Takens, Hyperbolicity & sensitive chaotic dynamics at homoclinic bifurcations, Cambridge Studies in Advanced Mathematics, 35, Cambridge University Press, Cambridge, 1993. | MR | Zbl

33. J. Palis, M. Viana, High dimension diffeomorphisms displaying infinitely many periodic attractors, Ann. Math., 140 (1994), 207-250 | MR | Zbl

34. S. Pilyugin, Shadowing in dynamical systems, Lect. Notes Math., vol. 1706, Springer, Berlin, 1999. | MR | Zbl

35. C. Pugh, The closing lemma, Amer. J. Math., 89 (1967), 956-1009 | MR | Zbl

36. C. Pugh, An improved closing lemma and a general density theorem, Amer. J. Math., 89 (1967), 1010-1021 | MR | Zbl

37. C. Pugh, C. Robinson, The C1-closing lemma, including hamiltonians, Ergodic Theory Dyn. Syst., 3 (1983), 261-314 | MR | Zbl

38. J. Robbin, A structural stability theorem, Ann. Math., 94 (1971), 447-493 | MR | Zbl

39. C. Robinson, Generic properties of conservative systems, Amer. J. Math., 92 (1970), 562-603 | MR | Zbl

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. | MR | Zbl

41. C. Robinson, Structural stability of C1-diffeomorphisms, J. Differ. Equ., 22 (1976), 28-73 | MR | Zbl

42. C. Robinson, Stability theorems and hyperbolicity in dynamical systems, Rocky Mt. J. Math., 7 (1977), 425-437 | MR | Zbl

43. N. Romero, Persistence of homoclinic tangencies in higher dimensions, Ergodic Theory Dyn. Syst., 15 (1995), 735-757 | MR | Zbl

44. K. Sakai, Diffeomorphisms with weak shadowing, Fundam. Math., 168 (2001), 57-75 | EuDML | MR | Zbl

45. M. Shub, Stability and genericity for diffeomorphisms, Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971), pp. 493-514, Academic Press, New York, 1973. | MR | Zbl

46. M. Shub, Topologically transitive diffeomorphisms of T4, Lect. Notes Math., vol. 206, pp. 39-40, Springer, Berlin-New York, 1971.

47. C. Simon, A 3-dimensional Abraham-Smale example, Proc. Amer. Math. Soc., 34 (1972), 629-630 | MR | Zbl

48. S. Smale, Stable manifolds for differential equations and diffeomorphisms, Ann. Sc. Norm. Super. Pisa, 17 (1963), 97-116 | EuDML | Numdam | MR | Zbl

49. S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817 | MR | Zbl

50. F. Takens, On Zeeman's tolerance stability conjecture, Lect. Notes Math., vol. 197, 209-219, Springer, Berlin, 1971. | Zbl

51. F. Takens, Tolerance stability, Lect. Notes Math., vol. 468, 293-304, Springer, Berlin, 1975. | MR | Zbl

52. L. Wen, A uniform C1 connecting lemma, Discrete Contin. Dyn. Syst., 8 (2002), 257-265 | MR | Zbl

53. L. Wen, Z. Xia, C1 connecting lemmas, Trans. Amer. Math. Soc., 352 (2000), 5213-5230 | MR | Zbl

54. W. White, On the tolerance stability conjecture, Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971), pp. 663-665, Academic Press, New York, 1973. | MR | Zbl

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 | MR | Zbl

Cité par Sources :