Abstract
LetM n be a compactn-dimensional manifold and ω be a symplectic or volume form onM n. Let ϕ be aC 1 diffeomorphism onM n that preserves ω and letp be a hyperbolic periodic point of Φ. We show that genericallyp has a homoclinic point, and moreover, the homoclinic points ofp is dense on both stable manifold and unstable manifold ofp. Takens [11] obtained the same result forn=2.
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References
Hayashi, S.: A connecting lemma. In Preparation
Newhouse, S.: Quasi-elliptic periodic points in conservative dynamical systems. Am. J. Math.99, 1061–87 (1977)
Oliveira, F.: On the generic existence of homoclinic points. Ergod. Th. & Dynam. Sys.7, 567–595 (1987)
Pixton, D.: Planar homoclinic points. J. Differ. Eqs.44, 365–382 (1982)
Poincare, H.: Les Methodes Nouvelles de la Mécanique Celeste. Tome II, 1899
Pugh, C.: The closing lemma. Am. J. Math.89, 956–1021 (1967)
Pugh, C.: TheC 1 connecting lemma. J. Dynam. & Diff. Eqs.4, No. 4, 545–553 (1992)
Pugh, C., Robinson, C.: TheC 1 closing lemma, including Hamiltonians. Ergod. Th. & Dynam. Sys.3, 261–313 (1983)
Robinson, C.: Generic Properties of Conservative Systems, I, II. Am. J. of Math.92, 562–603, 897–906 (1970)
Robinson, C.: Closing stable and unstable manifolds on the two-sphere. Proc. Am. Math. Soc.41, 299–303 (1973)
Takens, F.: Homoclinic points in conservative systems. Invent. Math.18, 267–292 (1972)
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Communicated by S.-T. Yau
Research supported in part by National Science Foundation.
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Xia, Z. Homoclinic points in symplectic and volume-preserving diffeomorphisms. Commun.Math. Phys. 177, 435–449 (1996). https://doi.org/10.1007/BF02101901
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DOI: https://doi.org/10.1007/BF02101901