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]
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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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},
}
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