Birth of homoclinic intersections: a model for the central dynamics of partially hyperbolic systems

Abstract

We prove a conjecture of J. Palis: Any diffeomorphism of a compact manifold can be $C^1$-approximated by a Morse-Smale diffeomorphism or by a diffeomorphism having a transverse homoclinic intersection.

  • [flavio] Go to document F. Abdenur, "Generic robustness of spectral decompositions," Ann. Sci. École Norm. Sup., vol. 36, iss. 2, pp. 213-224, 2003.
    @article {flavio, MRKEY = {1980311},
      AUTHOR = {Abdenur, Flavio},
      TITLE = {Generic robustness of spectral decompositions},
      JOURNAL = {Ann. Sci. École Norm. Sup.},
      FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
      VOLUME = {36},
      YEAR = {2003},
      NUMBER = {2},
      PAGES = {213--224},
      ISSN = {0012-9593},
      CODEN = {ASENAH},
      MRCLASS = {37C20 (37D20 37G25 57R50)},
      MRNUMBER = {2004b:37032},
      MRREVIEWER = {Boris Hasselblatt},
      DOI = {10.1016/S0012-9593(03)00008-9},
      ZBLNUMBER = {1027.37010},
      }
  • [abraham-smale] R. Abraham and S. Smale, "Nongenericity of $\Omega $-stability," in Global Analysis, I, Providence, R.I.: Amer. Math. Soc., 1970, vol. 14, pp. 5-8.
    @incollection {abraham-smale, MRKEY = {0271986},
      AUTHOR = {Abraham, R. and Smale, S.},
      TITLE = {Nongenericity of {$\Omega $}-stability},
      BOOKTITLE = {Global {A}nalysis, {\rm I}},
      SERUES={{P}roc. {S}ympos. {P}ure {M}ath.},
      VOLUME={14},
      PAGES = {5--8},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, R.I.},
      YEAR = {1970},
      MRCLASS = {57.50 (34.00)},
      MRNUMBER = {42 \#6867},
      MRREVIEWER = {J. W. Robbin},
      ZBLNUMBER = {0215.25102},
      }
  • [ABC] Go to document M. Arnaud, C. Bonatti, and S. Crovisier, "Dynamiques symplectiques génériques," Ergodic Theory Dynam. Systems, vol. 25, iss. 5, pp. 1401-1436, 2005.
    @article {ABC, MRKEY = {2173426},
      AUTHOR = {Arnaud, Marie-Claude and Bonatti, Christian and Crovisier, Sylvain},
      TITLE = {Dynamiques symplectiques génériques},
      JOURNAL = {Ergodic Theory Dynam. Systems},
      FJOURNAL = {Ergodic Theory and Dynamical Systems},
      VOLUME = {25},
      YEAR = {2005},
      NUMBER = {5},
      PAGES = {1401--1436},
      ISSN = {0143-3857},
      MRCLASS = {37J10 (37C05 37C20 37C50 37D20 37D30)},
      MRNUMBER = {2006k:37146},
      MRREVIEWER = {Lorenzo J. D{\'ı}az},
      DOI = {10.1017/S0143385704000975},
      ZBLNUMBER = {1084.37017},
      }
  • [birkhoff] G. D. Birkoff, "Nouvelles recherches sur les systèmes dynamiques," Memoriae Pont. Acad. Sci. Novi Lyncaei, vol. 1, pp. 85-216, 1935.
    @article{birkhoff,
      author = {Birkoff, G. D.},
      TITLE = {Nouvelles recherches sur les systèmes dynamiques},
      JOURNAL = {Memoriae Pont. Acad. Sci. Novi Lyncaei},
      VOLUME = {1},
      YEAR = {1935},
      PAGES = {85--216},
      NOTE = {and {\it Collected Math. Papers} {\bf II},
      530--659},
      ZBLNUMBER={0016.23401},
      }
  • [BC] Go to document C. Bonatti and S. Crovisier, "Récurrence et généricité," Invent. Math., vol. 158, iss. 1, pp. 33-104, 2004.
    @article {BC, MRKEY = {2090361},
      AUTHOR = {Bonatti, Christian and Crovisier, Sylvain},
      TITLE = {Récurrence et généricité},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {158},
      YEAR = {2004},
      NUMBER = {1},
      PAGES = {33--104},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {37C05 (37B20 37C20 37C29 37C50 37C70 37D05 37J10)},
      MRNUMBER = {2007b:37036},
      MRREVIEWER = {Lorenzo J. D{\'ı}az},
      DOI = {10.1007/s00222-004-0368-1},
      ZBLNUMBER = {1071.37015},
      }
  • [bdp] Go to document C. Bonatti, L. J. D’iaz, and E. R. Pujals, "A $C^1$-generic dichotomy for diffeomorphisms: weak forms of hyperbolicity or infinitely many sinks or sources," Ann. of Math., vol. 158, iss. 2, pp. 355-418, 2003.
    @article {bdp, MRKEY = {2018925},
      AUTHOR = {Bonatti, Christian and D{\'ı}az, L. J. and Pujals, E. R.},
      TITLE = {A {$C\sp 1$}-generic dichotomy for diffeomorphisms: weak forms of hyperbolicity or infinitely many sinks or sources},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {158},
      YEAR = {2003},
      NUMBER = {2},
      PAGES = {355--418},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {37D30 (37C20)},
      MRNUMBER = {2007k:37032},
      MRREVIEWER = {Maria Jos{é} Pacifico},
      DOI = {10.4007/annals.2003.158.355},
      ZBLNUMBER = {1049.37011},
      }
  • [bonatti-diaz-viana] C. Bonatti, L. J. D’iaz, and M. Viana, Dynamics Beyond Uniform Hyperbolicity, A Global Geometric and Probabilistic Perspective, Mathematical Physics, III, New York: Springer-Verlag, 2005, vol. 102.
    @book {bonatti-diaz-viana, MRKEY = {2105774},
      AUTHOR = {Bonatti, Christian and D{\'ı}az, Lorenzo J. and Viana, Marcelo},
      TITLE = {Dynamics Beyond Uniform Hyperbolicity, A Global Geometric and Probabilistic Perspective, Mathematical Physics, {\rm III}},
      SERIES = {Encyclopaedia Math. Sci.},
      VOLUME= {102},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {2005},
      PAGES = {xviii+384},
      ISBN = {3-540-22066-6},
      MRCLASS = {37-02 (37C20 37C29 37D25 37D30)},
      MRNUMBER = {2005g:37001},
      MRREVIEWER = {Sheldon E. Newhouse},
      ZBLNUMBER = {1060.37020},
      }
  • [bonatti-gan-wen] Go to document C. Bonatti, S. Gan, and L. Wen, "On the existence of non-trivial homoclinic classes," Ergodic Theory Dynam. Systems, vol. 27, iss. 5, pp. 1473-1508, 2007.
    @article {bonatti-gan-wen, MRKEY = {2358974},
      AUTHOR = {Bonatti, Christian and Gan, Shaobo and Wen, Lan},
      TITLE = {On the existence of non-trivial homoclinic classes},
      JOURNAL = {Ergodic Theory Dynam. Systems},
      FJOURNAL = {Ergodic Theory and Dynamical Systems},
      VOLUME = {27},
      YEAR = {2007},
      NUMBER = {5},
      PAGES = {1473--1508},
      ISSN = {0143-3857},
      MRCLASS = {37C20 (37C29 37D15 37D30 37D45 37E99)},
      MRNUMBER = {2009d:37036},
      MRREVIEWER = {Lorenzo J. D{\'ı}az},
      DOI = {10.1017/S0143385707000090},
      ZBLNUMBER = {1128.37021},
      }
  • [brin-pesin] M. Brin and Y. Pesin, "Partially hyperbolic dynamical systems," Math. USSR Izvestija, vol. 8, pp. 177-218, 1974.
    @article{brin-pesin,
      author = {Brin, M. and Pesin, Y.},
      TITLE = {Partially hyperbolic dynamical systems},
      JOURNAL = {Math. USSR Izvestija},
      VOLUME = {8},
      YEAR = {1974},
      PAGES = {177--218},
      ZBLCOMMENT = {BIBPROC: YEAR doesn't match found ZBLNUMBER},
      ZBLNUMBER = {0309.58017},
      }
  • [burns-wilkinson] K. Burns and A. Wilkinson, "On the ergodicity of partially hyperbolic systems," Ann. of Math., vol. 171, pp. 451-489, 2010.
    @article{burns-wilkinson,
      author = {Burns, K. and Wilkinson, A.},
      TITLE = {On the ergodicity of partially hyperbolic systems},
      JOURNAL = {Ann. of Math.},
      VOLUME = {171},
      YEAR={2010},
      PAGES={451--489},
      MRNUMBER={2630044},
      ZBLNUMBER={pre05712732},
      }
  • [conley] C. Conley, Isolated Invariant Sets and the Morse Index, Providence, R.I.: Amer. Math. Soc., 1978, vol. 38.
    @book {conley, MRKEY = {511133},
      AUTHOR = {Conley, Charles},
      TITLE = {Isolated Invariant Sets and the {M}orse Index},
      SERIES = {CBMS Reg. Conf. Ser. Math.},
      VOLUME = {38},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, R.I.},
      YEAR = {1978},
      PAGES = {iii+89},
      ISBN = {0-8218-1688-8},
      MRCLASS = {58Fxx (34-02 35-02)},
      MRNUMBER = {80c:58009},
      MRREVIEWER = {J. Smoller},
      ZBLNUMBER = {0397.34056},
      }
  • [crovisier] Go to document S. Crovisier, "Periodic orbits and chain-transitive sets of $C^1$-diffeomorphisms," Publ. Math. Inst. Hautes Études Sci., iss. 104, pp. 87-141, 2006.
    @article {crovisier, MRKEY = {2264835},
      AUTHOR = {Crovisier, Sylvain},
      TITLE = {Periodic orbits and chain-transitive sets of {$C\sp 1$}-diffeomorphisms},
      JOURNAL = {Publ. Math. Inst. Hautes Études Sci.},
      FJOURNAL = {Publications Mathématiques. Institut de Hautes Études Scientifiques},
      NUMBER = {104},
      YEAR = {2006},
      PAGES = {87--141},
      ISSN = {0073-8301},
      MRCLASS = {37C25 (37C05 37C20 37C50)},
      MRNUMBER = {2008f:37047},
      MRREVIEWER = {Lorenzo J. D{\'ı}az},
      DOI = {10.1007/s10240-006-0002-4},
      ZBLNUMBER = {05117095},
      }
  • [modele] S. Crovisier, Partial hyperbolicity far from homoclinic bifurcations, 2008.
    @misc{modele,
      author = {Crovisier, Sylvain},
      TITLE = {Partial hyperbolicity far from homoclinic bifurcations},
      YEAR = {2008},
      NOTE = {preprint},
      ARXIV = {0809.4965},
      }
  • [cp] S. Crovisier and E. Pujals, Essential hyperbolicity versus homoclinic bifurcations.
    @misc{cp,
      author = {Crovisier, Sylvain and Pujals, E.},
      TITLE = {Essential hyperbolicity versus homoclinic bifurcations},
      NOTE = {in preparation},
      }
  • [gan-wen] Go to document S. Gan and L. Wen, "Heteroclinic cycles and homoclinic closures for generic diffeomorphisms," J. Dynam. Differential Equations, vol. 15, iss. 2-3, pp. 451-471, 2003.
    @article {gan-wen, MRKEY = {2046726},
      AUTHOR = {Gan, Shaobo and Wen, Lan},
      TITLE = {Heteroclinic cycles and homoclinic closures for generic diffeomorphisms},
      NOTE = {Special issue dedicated to Victor A. Pliss on the occasion of his 70th birthday},
      JOURNAL = {J. Dynam. Differential Equations},
      FJOURNAL = {Journal of Dynamics and Differential Equations},
      VOLUME = {15},
      YEAR = {2003},
      NUMBER = {2-3},
      PAGES = {451--471},
      ISSN = {1040-7294},
      CODEN = {JDDEEH},
      MRCLASS = {37C05 (37C20 37C29)},
      MRNUMBER = {2005b:37029},
      MRREVIEWER = {Flavio Abdenur},
      DOI = {10.1023/B:JODY.0000009743.10365.9d},
      ZBLNUMBER = {1034.37013},
      }
  • [hayashi] S. Hayashi, "Connecting invariant manifolds and the solution of the $C^1$-stability and $\Omega$-stability conjectures for flows," Ann. of Math., vol. 145, pp. 81-137, 1997.
    @article{hayashi,
      author = {Hayashi, S.},
      TITLE = {Connecting invariant manifolds and the solution of the $C^1$-stability and $\Omega$-stability conjectures for flows},
      JOURNAL = {Ann. of Math.},
      VOLUME = {145},
      YEAR = {1997},
      PAGES = {81--137},
      NOTE = {and \textit{Ann. of Math.} {\bf 150} (1999), 353--356},
      ZBLNUMBER = {0871.58067},
      }
  • [HPS] M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, New York: Springer-Verlag, 1977, vol. 583.
    @book {HPS, MRKEY = {0501173},
      AUTHOR = {Hirsch, M. W. and Pugh, C. C. and Shub, M.},
      TITLE = {Invariant manifolds},
      SERIES = {Lecture Notes Math.},
      VOLUME={583},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1977},
      PAGES = {ii+149},
      MRCLASS = {58F15 (58F10)},
      MRNUMBER = {58 \#18595},
      MRREVIEWER = {M. C. Irwin},
      ZBLNUMBER = {0355.58009},
      }
  • [katok] Go to document A. Katok, "Lyapunov exponents, entropy and periodic orbits for diffeomorphisms," Inst. Hautes Études Sci. Publ. Math., iss. 51, pp. 137-173, 1980.
    @article {katok, MRKEY = {573822},
      AUTHOR = {Katok, A.},
      TITLE = {Lyapunov exponents, entropy and periodic orbits for diffeomorphisms},
      JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
      FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
      NUMBER = {51},
      YEAR = {1980},
      PAGES = {137--173},
      ISSN = {0073-8301},
      CODEN = {PMIHA6},
      MRCLASS = {28D20 (58F11 58F15)},
      MRNUMBER = {81i:28022},
      MRREVIEWER = {R. L. Adler},
      URL = {http://www.numdam.org/item?id=PMIHES_1980__51__137_0},
      ZBLNUMBER = {0445.58015},
      }
  • [kupka] I. Kupka, "Contribution à la théorie des champs génériques," Contributions to Differential Equations, vol. 2, pp. 457-484, 1963.
    @article {kupka, MRKEY = {0165536},
      AUTHOR = {Kupka, Ivan},
      TITLE = {Contribution à la théorie des champs génériques},
      JOURNAL = {Contributions to Differential Equations},
      VOLUME = {2},
      YEAR = {1963},
      PAGES = {457--484},
      MRCLASS = {57.48 (57.34)},
      MRNUMBER = {29 \#2818a},
      MRREVIEWER = {M. M. Peixoto},
      ZBLNUMBER = {0149.41002},
      }
  • [liao] S. T. Liao, "On the stability conjecture," Chinese Ann. Math., vol. 1, iss. 1, pp. 9-30, 1980.
    @article {liao, MRKEY = {591229},
      AUTHOR = {Liao, Shan Tao},
      TITLE = {On the stability conjecture},
      JOURNAL = {Chinese Ann. Math.},
      VOLUME = {1},
      YEAR = {1980},
      NUMBER = {1},
      PAGES = {9--30},
      MRCLASS = {58F10},
      MRNUMBER = {82c:58031},
      MRREVIEWER = {Clark Robinson},
      ZBLNUMBER = {0449.58013},
      }
  • [liao2] S. T. Liao, "Obstruction sets. II," Beijing Daxue Xuebao, iss. 2, pp. 1-36, 1981.
    @article {liao2, MRKEY = {646519},
      AUTHOR = {Liao, Shan Tao},
      TITLE = {Obstruction sets. {II}},
      JOURNAL = {Beijing Daxue Xuebao},
      FJOURNAL = {Acta Scientiarum Naturalium Universitatis Pekinensis},
      YEAR = {1981},
      NUMBER = {2},
      PAGES = {1--36},
      MRCLASS = {58F15 (58F99)},
      MRNUMBER = {83c:58065},
      MRREVIEWER = {Yieh Hei Wan},
      }
  • [mane] Go to document R. Mañé, "An ergodic closing lemma," Ann. of Math., vol. 116, iss. 3, pp. 503-540, 1982.
    @article {mane, MRKEY = {678479},
      AUTHOR = {Ma{ñ}{é},
      Ricardo},
      TITLE = {An ergodic closing lemma},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {116},
      YEAR = {1982},
      NUMBER = {3},
      PAGES = {503--540},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {58F10 (28D10 58F15)},
      MRNUMBER = {84f:58070},
      MRREVIEWER = {Carmen Chicone},
      DOI = {10.2307/2007021},
      ZBLNUMBER = {0511.58029},
      }
  • [newhouse-thesis] S. E. Newhouse, "Nondensity of axiom ${ A}({ a})$ on $S^{2}$," in Global Analysis, Providence, R.I.: Amer. Math. Soc., 1970, vol. 14, pp. 191-202.
    @incollection {newhouse-thesis, MRKEY = {0277005},
      AUTHOR = {Newhouse, Sheldon E.},
      TITLE = {Nondensity of axiom {${\rm A}({\rm a})$} on {$S\sp{2}$}},
      BOOKTITLE = {Global {A}nalysis},
      SERIES={{P}roc. {S}ympos. {P}ure {M}ath.},
      VOLUME={14},
      PAGES = {191--202},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, R.I.},
      YEAR = {1970},
      MRCLASS = {57.48},
      MRNUMBER = {43 \#2742},
      MRREVIEWER = {W. R. Utz},
      ZBLNUMBER = {0206.25801},
      }
  • [newhouse-homocline] S. E. Newhouse, "Lectures on dynamical systems," in Dynamical Systems, Mass.: Birkhäuser, 1980, vol. 8, pp. 1-114.
    @incollection {newhouse-homocline, MRKEY = {589590},
      AUTHOR = {Newhouse, Sheldon E.},
      TITLE = {Lectures on dynamical systems},
      BOOKTITLE = {Dynamical Systems},
      VENUE={{C}.{I}.{M}.{E}. {S}ummer {S}chool, {B}ressanone, 1978},
      SERIES = {Progr. Math.},
      VOLUME = {8},
      PAGES = {1--114},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Mass.},
      YEAR = {1980},
      MRCLASS = {58Fxx (34C35 70G99)},
      MRNUMBER = {81m:58028},
      MRREVIEWER = {M. I. Brin},
      ZBLNUMBER = {0444.58001},
      }
  • [oliveira1] Go to document F. Oliveira, "On the generic existence of homoclinic points," Ergodic Theory Dynam. Systems, vol. 7, iss. 4, pp. 567-595, 1987.
    @article {oliveira1, MRKEY = {922366},
      AUTHOR = {Oliveira, Fernando},
      TITLE = {On the generic existence of homoclinic points},
      JOURNAL = {Ergodic Theory Dynam. Systems},
      FJOURNAL = {Ergodic Theory and Dynamical Systems},
      VOLUME = {7},
      YEAR = {1987},
      NUMBER = {4},
      PAGES = {567--595},
      ISSN = {0143-3857},
      MRCLASS = {58F15 (58F20)},
      MRNUMBER = {89j:58104},
      DOI = {10.1017/S0143385700004211},
      ZBLNUMBER = {0612.58027},
      }
  • [oliveira2] Go to document F. Oliveira, "On the $C^\infty$ genericity of homoclinic orbits," Nonlinearity, vol. 13, iss. 3, pp. 653-662, 2000.
    @article {oliveira2, MRKEY = {1758993},
      AUTHOR = {Oliveira, Fernando},
      TITLE = {On the {$C\sp \infty$} genericity of homoclinic orbits},
      JOURNAL = {Nonlinearity},
      FJOURNAL = {Nonlinearity},
      VOLUME = {13},
      YEAR = {2000},
      NUMBER = {3},
      PAGES = {653--662},
      ISSN = {0951-7715},
      CODEN = {NONLE5},
      MRCLASS = {37G20 (37B40 37C29 37E30)},
      MRNUMBER = {2001h:37106},
      MRREVIEWER = {S. Yu. Pilyugin},
      DOI = {10.1088/0951-7715/13/3/308},
      ZBLNUMBER = {0983.37058},
      }
  • [palis-conjecture] J. Palis, "A global view of dynamics and a conjecture on the denseness of finitude of attractors," in Géométrie Complexe et Systèmes Dynamiques, Flexor, M., Ed., Paris: , 2000, vol. 261, pp. 335-347.
    @incollection{palis-conjecture, MRKEY = {1755446},
      AUTHOR = {Palis, Jacob},
      TITLE = {A global view of dynamics and a conjecture on the denseness of finitude of attractors},
      BOOKTITLE={G{é}om{é}trie Complexe et Syst{è}mes Dynamiques},
      VENUE={Orsay, France, July 3--8, 1995},
      SERIES = {Astérisque},
      EDITOR={Flexor, M.},
      ADDRESS={Paris},
      PAGES={335--347},
      FJOURNAL = {Astérisque},
      VOLUME = {261},
      YEAR = {2000},
      ISSN = {0303-1179},
      MRCLASS = {37C70 (37C20 37C75 37D45 37E30 37G25)},
      MRNUMBER = {2001d:37025},
      MRREVIEWER = {Rafael Labarca},
      ZBLNUMBER={1044.3704},
      }
  • [palis-conjecture2] Go to document J. Palis, "A global perspective for non-conservative dynamics," Ann. Inst. H. Poincaré Anal. Non Linéaire, vol. 22, iss. 4, pp. 485-507, 2005.
    @article {palis-conjecture2, MRKEY = {2145722},
      AUTHOR = {Palis, Jacob},
      TITLE = {A global perspective for non-conservative dynamics},
      JOURNAL = {Ann. Inst. H. Poincaré Anal. Non Linéaire},
      FJOURNAL = {Annales de l'Institut Henri Poincaré. Analyse Non Linéaire},
      VOLUME = {22},
      YEAR = {2005},
      NUMBER = {4},
      PAGES = {485--507},
      ISSN = {0294-1449},
      MRCLASS = {37C20 (37C29 37C40 37C70 37D45)},
      MRNUMBER = {2006b:37037},
      MRREVIEWER = {Leonardo E. Mora},
      DOI = {10.1016/j.anihpc.2005.01.001},
      ZBLNUMBER = {1143.37016},
      }
  • [palis-smale] J. Palis and S. Smale, "Structural stability theorems," in Global Analysis, Providence, R.I.: Amer. Math. Soc., 1970, vol. 14, pp. 223-231.
    @incollection {palis-smale, MRKEY = {0267603},
      AUTHOR = {Palis, Jacob and Smale, S.},
      TITLE = {Structural stability theorems},
      BOOKTITLE = {Global {A}nalysis},
      SERIES={{P}roc. {S}ympos. {P}ure {M}ath.},
      VOLUME={14},
      PAGES = {223--231},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, R.I.},
      YEAR = {1970},
      MRCLASS = {57.50},
      MRNUMBER = {42 \#2505},
      MRREVIEWER = {K. R. Meyer},
      ZBLNUMBER = {0214.50702},
      }
  • [palis-takens] J. Palis and F. Takens, Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations, Cambridge: Cambridge Univ. Press, 1993.
    @book {palis-takens, MRKEY = {1237641},
      AUTHOR = {Palis, Jacob and Takens, Floris},
      TITLE = {Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations},
      SERIES = {Cambridge Stud. Adv. Math.},
      NUMBER = {35},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {1993},
      PAGES = {x+234},
      ISBN = {0-521-39064-8},
      MRCLASS = {58F14 (58F13 58F15)},
      MRNUMBER = {94h:58129},
      MRREVIEWER = {Michael Hurley},
      ZBLNUMBER = {0790.58014},
      }
  • [peixoto] Go to document M. M. Peixoto, "Structural stability on two-dimensional manifolds," Topology, vol. 1, pp. 101-120, 1962.
    @article {peixoto, MRKEY = {0142859},
      AUTHOR = {Peixoto, M. M.},
      TITLE = {Structural stability on two-dimensional manifolds},
      JOURNAL = {Topology},
      FJOURNAL = {Topology. An International Journal of Mathematics},
      VOLUME = {1},
      YEAR = {1962},
      PAGES = {101--120},
      ISSN = {0040-9383},
      MRCLASS = {57.48 (57.50)},
      MRNUMBER = {26 \#426},
      MRREVIEWER = {J. K. Hale},
      DOI = {10.1016/0040-9383(65)90018-2},
      ZBLNUMBER = {0107.07103},
      }
  • [pixton] Go to document D. Pixton, "Planar homoclinic points," J. Differential Equations, vol. 44, iss. 3, pp. 365-382, 1982.
    @article {pixton, MRKEY = {661158},
      AUTHOR = {Pixton, Dennis},
      TITLE = {Planar homoclinic points},
      JOURNAL = {J. Differential Equations},
      FJOURNAL = {Journal of Differential Equations},
      VOLUME = {44},
      YEAR = {1982},
      NUMBER = {3},
      PAGES = {365--382},
      ISSN = {0022-0396},
      CODEN = {JDEQAK},
      MRCLASS = {58F15},
      MRNUMBER = {83h:58077},
      MRREVIEWER = {Michael J. Field},
      DOI = {10.1016/0022-0396(82)90002-X},
      ZBLNUMBER = {0506.58029},
      }
  • [poincare1] H. Poincaré, "Sur le problème des trois corps et les équations de la dynamique," Acta Math., vol. 13, pp. 1-270, 1890.
    @article{poincare1,
      author = {Poincaré, H.},
      TITLE = {Sur le problème des trois corps et les équations de la dynamique},
      JOURNAL = {Acta Math.},
      VOLUME = {13},
      YEAR = {1890},
      PAGES = {1--270},
      JFMNUMBER={22.0907.01},
      }
  • [poincare2] H. Poincaré, Les Méthodes Nouvelles de la Mécanique Céleste. Tome III, Gauthier-Villars, 1899.
    @book{poincare2,
      author = {Poincaré, H.},
      TITLE = {Les Méthodes Nouvelles de la Mécanique Céleste. Tome {\rm III}},
      PUBLISHER = {Gauthier-Villars},
      YEAR = {1899},
      JFMNUMBER={30.0834.08},
      }
  • [pugh] Go to document C. C. Pugh, "The closing lemma," Amer. J. Math., vol. 89, pp. 956-1009, 1967.
    @article {pugh, MRKEY = {0226669},
      AUTHOR = {Pugh, Charles C.},
      TITLE = {The closing lemma},
      JOURNAL = {Amer. J. Math.},
      FJOURNAL = {American Journal of Mathematics},
      VOLUME = {89},
      YEAR = {1967},
      PAGES = {956--1009},
      ISSN = {0002-9327},
      MRCLASS = {57.36 (34.00)},
      MRNUMBER = {37 \#2256},
      MRREVIEWER = {J. W. Robbin},
      DOI = {10.2307/2373413},
      ZBLNUMBER = {0167.21803},
      }
  • [pugh-robinson] Go to document C. C. Pugh and C. Robinson, "The $C^{1}$ closing lemma, including Hamiltonians," Ergodic Theory Dynam. Systems, vol. 3, iss. 2, pp. 261-313, 1983.
    @article {pugh-robinson, MRKEY = {742228},
      AUTHOR = {Pugh, Charles C. and Robinson, Clark},
      TITLE = {The {$C\sp{1}$} closing lemma, including {H}amiltonians},
      JOURNAL = {Ergodic Theory Dynam. Systems},
      FJOURNAL = {Ergodic Theory and Dynamical Systems},
      VOLUME = {3},
      YEAR = {1983},
      NUMBER = {2},
      PAGES = {261--313},
      ISSN = {0143-3857},
      MRCLASS = {58F10 (58F30)},
      MRNUMBER = {85m:58106},
      MRREVIEWER = {Richard Cushman},
      DOI = {10.1017/S0143385700001978},
      ZBLNUMBER = {0548.58012},
      }
  • [pujals1] Go to document E. R. Pujals, "On the density of hyperbolicity and homoclinic bifurcations for 3D-diffeomorphisms in attracting regions," Discrete Contin. Dyn. Syst., vol. 16, iss. 1, pp. 179-226, 2006.
    @article {pujals1, MRKEY = {2221742},
      AUTHOR = {Pujals, Enrique R.},
      TITLE = {On the density of hyperbolicity and homoclinic bifurcations for 3{D}-diffeomorphisms in attracting regions},
      JOURNAL = {Discrete Contin. Dyn. Syst.},
      FJOURNAL = {Discrete and Continuous Dynamical Systems. Series A},
      VOLUME = {16},
      YEAR = {2006},
      NUMBER = {1},
      PAGES = {179--226},
      ISSN = {1078-0947},
      MRCLASS = {37C20 (37C29 37D05 37E99 37G25)},
      MRNUMBER = {2007d:37018},
      MRREVIEWER = {Slobodan N. Simi{ć}},
      DOI = {10.3934/dcds.2006.16.179},
      ZBLNUMBER = {1111.37008},
      }
  • [pujals2] Go to document E. R. Pujals, "Density of hyperbolicity and homoclinic bifurcations for attracting topologically hyperbolic sets," Discrete Contin. Dyn. Syst., vol. 20, iss. 2, pp. 335-405, 2008.
    @article {pujals2, MRKEY = {2358263},
      AUTHOR = {Pujals, Enrique R.},
      TITLE = {Density of hyperbolicity and homoclinic bifurcations for attracting topologically hyperbolic sets},
      JOURNAL = {Discrete Contin. Dyn. Syst.},
      FJOURNAL = {Discrete and Continuous Dynamical Systems. Series A},
      VOLUME = {20},
      YEAR = {2008},
      NUMBER = {2},
      PAGES = {335--405},
      ISSN = {1078-0947},
      MRCLASS = {37D05 (37C70 37D30 37G25)},
      MRNUMBER = {2008j:37056},
      MRREVIEWER = {Leonardo E. Mora},
      DOI = {10.3934/dcds.2008.20.335},
      ZBLNUMBER = {1144.37008},
      }
  • [pujals-sambarino] Go to document E. R. Pujals and M. Sambarino, "Homoclinic tangencies and hyperbolicity for surface diffeomorphisms," Ann. of Math., vol. 151, iss. 3, pp. 961-1023, 2000.
    @article {pujals-sambarino, MRKEY = {1779562},
      AUTHOR = {Pujals, Enrique R. and Sambarino, Martin},
      TITLE = {Homoclinic tangencies and hyperbolicity for surface diffeomorphisms},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {151},
      YEAR = {2000},
      NUMBER = {3},
      PAGES = {961--1023},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {37D20 (37C29 37C70 37D15 37E30)},
      MRNUMBER = {2001m:37057},
      MRREVIEWER = {Yu. S. Il{\cprime}yashenko},
      DOI = {10.2307/121127},
      ZBLNUMBER = {0959.37040},
      }
  • [pujals-sambarino-codimension] Go to document E. R. Pujals and M. Sambarino, "Density of hyperbolicity and tangencies in sectional dissipative regions," Ann. Inst. H. Poincaré Anal. Non Linéaire, vol. 26, iss. 5, pp. 1971-2000, 2009.
    @article {pujals-sambarino-codimension, MRKEY = {2566718},
      AUTHOR = {Pujals, Enrique R. and Sambarino, Martin},
      TITLE = {Density of hyperbolicity and tangencies in sectional dissipative regions},
      JOURNAL = {Ann. Inst. H. Poincaré Anal. Non Linéaire},
      FJOURNAL = {Annales de l'Institut Henri Poincaré. Analyse Non Linéaire},
      VOLUME = {26},
      YEAR = {2009},
      NUMBER = {5},
      PAGES = {1971--2000},
      ISSN = {0294-1449},
      MRCLASS = {37Dxx (37Cxx)},
      MRNUMBER = {2566718},
      DOI = {10.1016/j.anihpc.2009.04.003},
      ZBLNUMBER = {1189.37032},
      }
  • [robinson-periodic] Go to document C. Robinson, "Generic properties of conservative systems," Amer. J. Math., vol. 92, pp. 562-603, 1970.
    @article {robinson-periodic, MRKEY = {0273640},
      AUTHOR = {Robinson, Clark},
      TITLE = {Generic properties of conservative systems},
      JOURNAL = {Amer. J. Math.},
      FJOURNAL = {American Journal of Mathematics},
      VOLUME = {92},
      YEAR = {1970},
      PAGES = {562--603},
      ISSN = {0002-9327},
      MRCLASS = {57.34 (34.00)},
      MRNUMBER = {42 \#8517},
      MRREVIEWER = {F. W. Wilson, Jr.},
      DOI = {10.2307/2373361},
      ZBLNUMBER = {0212.56502},
      }
  • [robinson-homocline] Go to document C. Robinson, "Closing stable and unstable manifolds on the two sphere," Proc. Amer. Math. Soc., vol. 41, pp. 299-303, 1973.
    @article {robinson-homocline, MRKEY = {0321141},
      AUTHOR = {Robinson, Clark},
      TITLE = {Closing stable and unstable manifolds on the two sphere},
      JOURNAL = {Proc. Amer. Math. Soc.},
      FJOURNAL = {Proceedings of the American Mathematical Society},
      VOLUME = {41},
      YEAR = {1973},
      PAGES = {299--303},
      ISSN = {0002-9939},
      MRCLASS = {58F99},
      MRNUMBER = {47 \#9674},
      MRREVIEWER = {J. W. Robbin},
      DOI = {10.2307/2038860},
      ZBLNUMBER = {0275.58013},
      }
  • [robinson-livre] C. Robinson, Dynamical Systems. Stability, Symbolic Dynamics, and Chaos, Boca Raton, FL: CRC Press, 1995.
    @book {robinson-livre, MRKEY = {1396532},
      AUTHOR = {Robinson, Clark},
      TITLE = {Dynamical Systems. Stability, Symbolic Dynamics, and Chaos},
      SERIES = {Studies Adv. Math.},
      PUBLISHER = {CRC Press},
      ADDRESS = {Boca Raton, FL},
      YEAR = {1995},
      PAGES = {xii+468},
      ISBN = {0-8493-8493-1},
      MRCLASS = {58Fxx (34C35 58-01)},
      MRNUMBER = {97e:58064},
      MRREVIEWER = {Michael Hurley},
      ZBLNUMBER = {0853.58001},
      }
  • [smale-morse] Go to document S. Smale, "Morse inequalities for a dynamical system," Bull. Amer. Math. Soc., vol. 66, pp. 43-49, 1960.
    @article {smale-morse, MRKEY = {0117745},
      AUTHOR = {Smale, Stephen},
      TITLE = {Morse inequalities for a dynamical system},
      JOURNAL = {Bull. Amer. Math. Soc.},
      FJOURNAL = {Bulletin of the American Mathematical Society},
      VOLUME = {66},
      YEAR = {1960},
      PAGES = {43--49},
      ISSN = {0002-9904},
      MRCLASS = {57.00},
      MRNUMBER = {22 \#8519},
      MRREVIEWER = {M. M. Peixoto},
      DOI = {10.1090/S0002-9904-1960-10386-2},
      ZBLNUMBER = {0100.29701},
      }
  • [smale-kupka] S. Smale, "Stable manifolds for differential equations and diffeomorphisms," Ann. Scuola Norm. Sup. Pisa, vol. 17, pp. 97-116, 1963.
    @article {smale-kupka, MRKEY = {0165537},
      AUTHOR = {Smale, Stephen},
      TITLE = {Stable manifolds for differential equations and diffeomorphisms},
      JOURNAL = {Ann. Scuola Norm. Sup. Pisa},
      VOLUME = {17},
      YEAR = {1963},
      PAGES = {97--116},
      MRCLASS = {57.48 (57.34)},
      MRNUMBER = {29 \#2818b},
      MRREVIEWER = {M. M. Peixoto},
      ZBLNUMBER = {0113.29702},
      }
  • [smale-horseshoe] S. Smale, "Diffeomorphisms with many periodic points," in Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton, N.J.: Princeton Univ. Press, 1965, pp. 63-80.
    @incollection {smale-horseshoe, MRKEY = {0182020},
      AUTHOR = {Smale, Stephen},
      TITLE = {Diffeomorphisms with many periodic points},
      BOOKTITLE = {Differential and {C}ombinatorial {T}opology ({A} {S}ymposium in {H}onor of {M}arston {M}orse)},
      PAGES = {63--80},
      PUBLISHER = {Princeton Univ. Press},
      ADDRESS = {Princeton, N.J.},
      YEAR = {1965},
      MRCLASS = {57.47},
      MRNUMBER = {31 \#6244},
      MRREVIEWER = {M. F. Atiyah},
      ZBLNUMBER = {0142.41103},
      }
  • [smale-dynamics] Go to document S. Smale, "Differentiable dynamical systems," Bull. Amer. Math. Soc., vol. 73, pp. 747-817, 1967.
    @article {smale-dynamics, MRKEY = {0228014},
      AUTHOR = {Smale, Stephen},
      TITLE = {Differentiable dynamical systems},
      JOURNAL = {Bull. Amer. Math. Soc.},
      FJOURNAL = {Bulletin of the American Mathematical Society},
      VOLUME = {73},
      YEAR = {1967},
      PAGES = {747--817},
      ISSN = {0002-9904},
      MRCLASS = {57.47 (34.00)},
      MRNUMBER = {37 \#3598},
      MRREVIEWER = {M. M. Peixoto},
      DOI = {10.1090/S0002-9904-1967-11798-1},
      ZBLNUMBER = {0202.55202},
      }
  • [takens] Go to document F. Takens, "Homoclinic points in conservative systems," Invent. Math., vol. 18, pp. 267-292, 1972.
    @article {takens, MRKEY = {0331435},
      AUTHOR = {Takens, Floris},
      TITLE = {Homoclinic points in conservative systems},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {18},
      YEAR = {1972},
      PAGES = {267--292},
      ISSN = {0020-9910},
      MRCLASS = {58F20 (58F10)},
      MRNUMBER = {48 \#9768},
      MRREVIEWER = {Clark Robinson},
      DOI = {10.1007/BF01389816},
      ZBLNUMBER = {0247.58007},
      }
  • [wen] Go to document L. Wen, "Generic diffeomorphisms away from homoclinic tangencies and heterodimensional cycles," Bull. Braz. Math. Soc., vol. 35, iss. 3, pp. 419-452, 2004.
    @article {wen, MRKEY = {2106314},
      AUTHOR = {Wen, Lan},
      TITLE = {Generic diffeomorphisms away from homoclinic tangencies and heterodimensional cycles},
      JOURNAL = {Bull. Braz. Math. Soc.},
      FJOURNAL = {Bulletin of the Brazilian Mathematical Society. New Series. Boletim da Sociedade Brasileira de Matemática},
      VOLUME = {35},
      YEAR = {2004},
      NUMBER = {3},
      PAGES = {419--452},
      ISSN = {1678-7544},
      MRCLASS = {37D30 (37C20 37G25)},
      MRNUMBER = {2005i:37035},
      MRREVIEWER = {Flavio Abdenur},
      DOI = {10.1007/s00574-004-0023-x},
      ZBLNUMBER = {1099.37024},
      }
  • [xia] Go to document Z. Xia, "Homoclinic points in symplectic and volume-preserving diffeomorphisms," Comm. Math. Phys., vol. 177, iss. 2, pp. 435-449, 1996.
    @article {xia, MRKEY = {1384143},
      AUTHOR = {Xia, Zhihong},
      TITLE = {Homoclinic points in symplectic and volume-preserving diffeomorphisms},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {177},
      YEAR = {1996},
      NUMBER = {2},
      PAGES = {435--449},
      ISSN = {0010-3616},
      CODEN = {CMPHAY},
      MRCLASS = {58F15 (58F30)},
      MRNUMBER = {97d:58154},
      MRREVIEWER = {Marcelo Viana},
      URL = {http://projecteuclid.org/getRecord?id=euclid.cmp/1104286336},
      ZBLNUMBER = {0959.37050},
      }
  • [zehnder] Go to document E. Zehnder, "Homoclinic points near elliptic fixed points," Comm. Pure Appl. Math., vol. 26, pp. 131-182, 1973.
    @article {zehnder, MRKEY = {0345134},
      AUTHOR = {Zehnder, E.},
      TITLE = {Homoclinic points near elliptic fixed points},
      JOURNAL = {Comm. Pure Appl. Math.},
      FJOURNAL = {Communications on Pure and Applied Mathematics},
      VOLUME = {26},
      YEAR = {1973},
      PAGES = {131--182},
      ISSN = {0010-3640},
      MRCLASS = {58F05 (58F20)},
      MRNUMBER = {49 \#9873},
      MRREVIEWER = {N. Fenichel},
      DOI = {10.1002/cpa.3160260204},
      ZBLNUMBER = {0261.58002},
      }
  • [zhang] Y. Zhang, "Codimension one dominated splittings," Peking University, thesis , 2001.
    @techreport{zhang,
      author = {Zhang, Y.},
      TITLE = {Codimension one dominated splittings},
      TYPE = {thesis},
      INSTITUTION = {Peking University},
      YEAR = {2001},
      ZBLCOMMENT = {BIBPROC: YEAR doesn't match found ZBLNUMBER},
      ZBLNUMBER = {1085.37025},
      }

Authors

Sylvain Crovisier

CNRS - Laboratoire Analyse, Géométrie et Applications
UMR 7539
Institut Galilée, Université Paris 13
Avenue J.-B. Clément, 93430
Villetaneuse
France