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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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