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.
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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.
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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},
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[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},
}