Non-hyperbolic Closed Characteristics on Non-degenerate Star-shaped Hypersurfaces in R<sup>2<i>n</i></sup>

doi:10.1007/s10114-016-6019-9

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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.

Key words： Closed characteristic star-shaped hypersurface non-hyperbolic Maslov-type index

Received: 08 January 2016

MR (2010):	58E05
	37J45
	34C25

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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	Hua Gui DUAN
	Hui LIU
	Yi Ming LONG
	Wei WANG

Cite this article:

Hua Gui DUAN,Hui LIU,Yi Ming LONG等. Non-hyperbolic Closed Characteristics on Non-degenerate Star-shaped Hypersurfaces in R²ⁿ[J]. Acta Mathematica Sinica, English Series, 2018, 34(1): 1-18.

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http://123.57.41.99/Jwk_sxxb_en/EN/10.1007/s10114-016-6019-9 OR http://123.57.41.99/Jwk_sxxb_en/EN/Y2018/V34/I1/1

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