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Closed characteristics on non-degenerate star-shaped hypersurfaces in R2n

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Abstract

In this paper, firstly we establish the relation theorem between the Maslov-type index and the index defined by C. Viterbo for star-shaped Hamiltonian systems. Then we extend the iteration formula ofC. Viterbo for non-degenerate star-shaped Hamiltonian systems to the general case. Finally we prove that there exist at least two geometrically distinct closed characteristics on any non-degenerate star-shaped compact smooth hypersurface on R2n with n > 1. Here we call a hypersurface non-degenerate, if all the closed characteristics on the given hypersurface together with all of their iterations are non-degenerate as periodic solutions of the corresponding Hamiltonian system. We also study the ellipticity of closed characteristics whenn = 2

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Authors and Affiliations

  1. Nankai Institute of Mathematics, Nankai University, 300071, Tianjin, China

    Xijun Hu & Yiming Long

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  1. Xijun Hu
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  2. Yiming Long
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Correspondence to Yiming Long.

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Hu, X., Long, Y. Closed characteristics on non-degenerate star-shaped hypersurfaces in R2n . Sci. China Ser. A-Math. 45, 1038–1052 (2002). https://doi.org/10.1007/BF02879987

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  • DOI: https://doi.org/10.1007/BF02879987

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