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Homoclinic points in symplectic and volume-preserving diffeomorphisms. (English) Zbl 0959.37050
Summary: Let $$M^n$$ be a compact $$n$$-dimensional manifold and $$\omega$$ be a symplectic or volume form on $$M^n$$. Let $$\phi$$ be a $$C^1$$ diffeomorphism on $$M^n$$ that preserves $$\omega$$ and let $$p$$ be a hyperbolic periodic point of $$\phi$$. We show that generically $$p$$ has a homoclinic point, and moreover, the homoclinic points of $$p$$ are dense on both stable manifold and unstable manifold of $$p$$. F. Takens [Invent. Math. 18, 267-292 (1972; Zbl 0247.58007)] obtained the same result for $$n=2$$.

##### MSC:
 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
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##### References:
 [1] Hayashi, S.: A connecting lemma. In Preparation · Zbl 0843.58081 [2] Newhouse, S.: Quasi-elliptic periodic points in conservative dynamical systems. Am. J. Math.99, 1061–87 (1977) · Zbl 0379.58011 · doi:10.2307/2374000 [3] Oliveira, F.: On the generic existence of homoclinic points. Ergod. Th. & Dynam. Sys.7, 567–595 (1987) · Zbl 0612.58027 [4] Pixton, D.: Planar homoclinic points. J. Differ. Eqs.44, 365–382 (1982) · Zbl 0506.58029 · doi:10.1016/0022-0396(82)90002-X [5] Poincare, H.: Les Methodes Nouvelles de la Mécanique Celeste. Tome II, 1899 [6] Pugh, C.: The closing lemma. Am. J. Math.89, 956–1021 (1967) · Zbl 0167.21803 · doi:10.2307/2373413 [7] Pugh, C.: TheC 1 connecting lemma. J. Dynam. & Diff. Eqs.4, No. 4, 545–553 (1992) · Zbl 0764.58030 · doi:10.1007/BF01048259 [8] Pugh, C., Robinson, C.: TheC 1 closing lemma, including Hamiltonians. Ergod. Th. & Dynam. Sys.3, 261–313 (1983) · Zbl 0548.58012 [9] Robinson, C.: Generic Properties of Conservative Systems, I, II. Am. J. of Math.92, 562–603, 897–906 (1970) · Zbl 0212.56502 · doi:10.2307/2373361 [10] Robinson, C.: Closing stable and unstable manifolds on the two-sphere. Proc. Am. Math. Soc.41, 299–303 (1973) · Zbl 0275.58013 · doi:10.1090/S0002-9939-1973-0321141-7 [11] Takens, F.: Homoclinic points in conservative systems. Invent. Math.18, 267–292 (1972) · Zbl 0247.58007 · doi:10.1007/BF01389816
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