2002
DOI: 10.1007/s002089100257
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Multiplicity of closed characteristics on symmetric convex hypersurfaces in $\mathbb{R}^{2n}$

Abstract: Let Σ be a compact C 2 hypersurface in R 2n bounding a convex set with non-empty interior. In this paper it is proved that there always exist at least n geometrically distinct closed characteristics on Σ if Σ is symmetric with respect to the origin.

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Cited by 59 publications
(44 citation statements)
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“…These index theories (about the properties of the index) and their iteration theories have important applications in the study of nonlinear Hamiltonian systems. See [Fei 1995;Conley and Zehnder 1984;Long and Zehnder 1990;Fei and Qiu 1997;Chang et al 1997;Su 1998;Li and Liu 1989] for multiple periodic solutions of asymptotically linear Hamiltonian systems, [Ekeland and Hofer 1985;Dong and Long 1997] for Rabinowitz's minimal periodic problem, and [Ekeland and Hofer 1987;Long and Zhu 2002;Liu et al 2002] for multiple closed characteristics on compact convex hyper-surfaces in ‫ޒ‬ 2n . For a systematic treatment and other applications, one can refer to the excellent books [Ekeland 1990;Long 2002].…”
Section: Introductionmentioning
confidence: 99%
“…These index theories (about the properties of the index) and their iteration theories have important applications in the study of nonlinear Hamiltonian systems. See [Fei 1995;Conley and Zehnder 1984;Long and Zehnder 1990;Fei and Qiu 1997;Chang et al 1997;Su 1998;Li and Liu 1989] for multiple periodic solutions of asymptotically linear Hamiltonian systems, [Ekeland and Hofer 1985;Dong and Long 1997] for Rabinowitz's minimal periodic problem, and [Ekeland and Hofer 1987;Long and Zhu 2002;Liu et al 2002] for multiple closed characteristics on compact convex hyper-surfaces in ‫ޒ‬ 2n . For a systematic treatment and other applications, one can refer to the excellent books [Ekeland 1990;Long 2002].…”
Section: Introductionmentioning
confidence: 99%
“…There exists a huge amount of literature concerning the study of brake orbitssee e.g., [12,15,21,22] -and more generally on the study of periodic solutions of autonomous Hamiltonian systems with prescribed energy [13,14,16,17,18]. We also observe here that manifolds with singular boundary of the type investigated in the present paper arise naturally in the study of certain compactifications of incomplete Riemannian manifolds.…”
Section: Introductionmentioning
confidence: 51%
“…Since ∈ SH st (4), by Theorem 1 of [11] and Lemma 4.2 of [19], both (τ 1 , y 1 ) and (τ 2 , y 2 ) must be symmetric. Let ψ j = γ j | [0,τ j /2] , then we have γ j | [0,τ j ] = ψ 2 j and γ j (τ j ) = γ j (τ j /2) 2 .…”
Section: Proof Of Theorem 14 Letmentioning
confidence: 99%