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Non-hyperbolic Closed Characteristics on Non-degenerate Star-shaped Hypersurfaces in R2n |
Hua Gui DUAN1, Hui LIU2, Yi Ming LONG3, Wei WANG4 |
1. School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, P. R. China; 2. School of Mathematics and Statistics, Wuhan University, Wuhan 430072, P. R. China; 3. Chern Institute of Mathematics and LPMC, Nankai University Tianjin 300071, P. R. China; 4. School of Mathematical Sciences and LMAM, Peking University, Beijing 100871, P. R. China |
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Abstract In this paper, we prove that for every index perfect non-degenerate compact star-shaped hypersurface ∑ ⊂ R2n, there exist at least n non-hyperbolic closed characteristics with even Maslovtype indices on ∑ when n is even. When n is odd, there exist at least n closed characteristics with odd Maslov-type indices on ∑ and at least (n-1) of them are non-hyperbolic. Here we call a compact star-shaped hypersurface ∑ ⊂ R2n index perfect if it carries only finitely many geometrically distinct prime closed characteristics, and every prime closed characteristic (τ, y) on ∑ possesses positive mean index and whose Maslov-type index i(y, m) of its m-th iterate satisfies i(y, m) ≠-1 when n is even, and i(y, m) ∉ {-2, -1, 0} when n is odd for all m ∈ N.
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Received: 08 January 2016
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Fund:The first author is supported by NSFC (Grant Nos. 11671215, 11131004 and 11471169) and LPMC of MOE of China; the second author is supported by NSFC (Grant No. 11401555) and Anhui Provincial Natural Science Foundation (Grant No. 1608085QA01); the third author is supported by NSFC (Nos. 11131004 and 11671215), MCME, LPMC of MOE of China, Nankai University and BAICIT of Capital Normal University; the fourth author is supported by NSFC (Grant Nos. 11222105 and 11431001) |
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