On maximal transitive sets of generic diffeomorphisms

Christian Bonatti; Lorenzo J. Díaz

Publications Mathématiques de l'IHÉS (2003)

  • Volume: 96, page 171-197
  • ISSN: 0073-8301

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Bonatti, Christian, and Díaz, Lorenzo J.. "On maximal transitive sets of generic diffeomorphisms." Publications Mathématiques de l'IHÉS 96 (2003): 171-197. <http://eudml.org/doc/104185>.

@article{Bonatti2003,
author = {Bonatti, Christian, Díaz, Lorenzo J.},
journal = {Publications Mathématiques de l'IHÉS},
keywords = {locally generic diffeomorphisms; wild homoclinic classes; maximal transitive invariant sets; robustly transitive sets; Newhouse phenomenon},
language = {eng},
pages = {171-197},
publisher = {Institut des Hautes Etudes Scientifiques},
title = {On maximal transitive sets of generic diffeomorphisms},
url = {http://eudml.org/doc/104185},
volume = {96},
year = {2003},
}

TY - JOUR
AU - Bonatti, Christian
AU - Díaz, Lorenzo J.
TI - On maximal transitive sets of generic diffeomorphisms
JO - Publications Mathématiques de l'IHÉS
PY - 2003
PB - Institut des Hautes Etudes Scientifiques
VL - 96
SP - 171
EP - 197
LA - eng
KW - locally generic diffeomorphisms; wild homoclinic classes; maximal transitive invariant sets; robustly transitive sets; Newhouse phenomenon
UR - http://eudml.org/doc/104185
ER -

References

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  1. [Ab] F. ABDENUR, Generic robustness of a spectral decompositions, to appear in Ann. Scient. Ec. Norm. Sup.. Zbl1027.37010
  2. [AS] R. ABRAHAM and S. SMALE, Nongenericity of ω -stability, Global Analysis I, Proc. Symp. Pure Math. A.M.S., 14 (1968), 5-8. Zbl0215.25102MR271986
  3. [Ar] M.-C. ARNAUD, Creation de connexions en topologie C 1 , Ergod. Th. Dynam. Syst., 21 (2001), 339-381. Zbl0997.37007MR1827109
  4. [BD1] Ch. BONATTI and L. J. DÍAZ, Persistent nonhyperbolic transitive diffeomorphims, Ann. Math., 143 (1996), 357-396. Zbl0852.58066MR1381990
  5. [BD2] Ch. BONATTI and L. J. DÍAZ, Connexions hétéroclines et généricité d’une infinité de puits ou de sources, Ann. Scient. Ec. Norm. Sup., 4e série, t32 (1999), 135-150. Zbl0944.37012
  6. [BDP] Ch. BONATTI, L. J. DÍAZ and E. R. PUJALS, A C 1 -generic dichotomy for diffeomorphisms: Weak forms of hyperbolicicity or infinitely many sinks or sources, to appear in Annals of Math.. Zbl1049.37011MR2018925
  7. [BGLT] Ch. BONATTI, J.-M. GAMBAUDO, J.-M. LION and Ch. TRESSER, Wandering domains for infinitely renormalizable diffeomorphisms of the disk, Proc. Amer. Math. Soc., 122 (2) (1994), 1273-1278. Zbl0843.57016MR1223264
  8. [BS] J. BUESCU and I. STEWART, Liapunov stability and adding machines, Ergodic Th. Dynam. Syst., 15 (2) (1995),271-290. Zbl0848.54027MR1332404
  9. [CM] C. CARBALLO and C. MORALES, Homoclinic classes and finitude of attractors for vector fields in n-manifolds, reprint (2001). Zbl1035.37007
  10. [CMP] C. CARBALLO, C. MORALES and M. J. PACÍFICO, Homoclinic classes for C 1 -generic vector fields, to appear in Ergodic Th. Dynam. Syst.. Zbl1047.37009
  11. [Ce] J. CERF, Sur les difféomorphismes de la sphère de dimension trois ( Γ 4 = 0 ) , Lecture Notes in Mathematics 53 1968), pp. xii + 133, Springer-Verlag. Zbl0164.24502MR229250
  12. [DPU] L. J. DÍAZ, E. R. PUJALS and R. URES, Partial hyperbolicity and robust transitivity, Acta Math., 183 (1999), -43. Zbl0987.37020MR1719547
  13. [DR] L. J. DÍAZ and J. ROCHA, Partially hyperbolic and transitive dynamics generated by heteroclinic cycles, Ergodic Th. Dyn. Syst., 25 (2001), 25-76. Zbl0972.37018MR1826660
  14. [DS] L. J. DÍAZ and B. SANTORO, Collision, explosion and collapse of homoclinic classes, preprint (2002). Zbl1046.37035MR2057137
  15. [Fr] J. FRANKS, Necessary conditions for stability of diffeomorphisms, Trans. A.M.S., 158 (1971), 301-308. Zbl0219.58005MR283812
  16. [Fu] H. FURSTENBERG, Strict ergodicity and transformations of the torus, Amer. J. of Math., 83 (1961), 573-601. Zbl0178.38404MR133429
  17. [H] S. HAYASHI, Connecting invariant manifolds and the solution of the C 1 -stability and ω -stability conjectures for flows, Ann. of Math., 145 (1997), 81-137. Zbl0871.58067MR1432037
  18. [Hu] M. HURLEY, Attractors: persistence and density of their basins, Trans. Amer. Math. Soc., 269 (1982), 247-271. Zbl0494.58023MR637037
  19. [Ma] R. MANE, An ergodic closing lemma, Ann. of Math., 116 (1982), 503-540. Zbl0511.58029MR678479
  20. [Mi] J. W. MILNOR, Topology from differential view point, Charlottesville, The University Press of Virginia, 1965. 
  21. [N1] S. NEWHOUSE, Diffeomorphisms with infinitely many sinks, Topology, 13 (1974), 9-18. Zbl0275.58016MR339291
  22. [N2] S. NEWHOUSE, The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms, Publ. Math. IHES, 50 (1979), 101-151. Zbl0445.58022MR556584
  23. [PT] J. PALIS and F. TAKENS, Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations, Cambridge Studies in Advanced Mathematics, 35 (1993). Zbl0790.58014MR1237641
  24. [PV] J. PALIS and M. VIANA, High dimension diffeomorphisms displaying infinitely many sinks, Ann. of Math., 140 (1994), 207-250. Zbl0817.58004MR1289496
  25. [Pu] C. PUGH, The closing lemma, Amer. Jour. of Math., 89 (1967), 956-1009. Zbl0167.21803MR226669
  26. [Si] R. C. SIMON, A 3-dimensional Abraham-Smale example, Proc. A.M.S., 34 (2) (1972), 629-630. Zbl0259.58006MR295391
  27. [Sm] S. SMALE, Differentiable dynamical systems, Bull. A.M.S., 73 (1967), 747-817. Zbl0202.55202MR228014
  28. [Tj] J. C. TATJER, Three dimensional dissipative diffeomorphisms with homoclinic tangencies, Ergod. Th. Dynam. Syst., 21 (2001), 249-302. Zbl0972.37013MR1826668

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