Closed characteristics on compact convex hypersurfaces in $\R^{2n}$

By Y. Long, Chaofeng Zhu. 2001

211 citations recorded by the source corpus.

For any given compact C^2 hypersurface \Sigma in {\bf R}^{2n} bounding a strictly convex set with nonempty interior, in this paper an invariant \varrho_n(\Sigma) is defined and satisfies \varrho_n(\Sigma)\ge [n/2]+1, where [a] denotes the greatest integer which is not greater than a\in {\bf R}. The following results are proved in this paper. There always exist at least \rho_n(\Sigma) geometrically distinct closed characteristics on \Sigma. If all the geometrically distinct closed characteristics on \Sigma are nondegenerate, then \varrho_n(\Sigma)\ge n. If the total number of geometrically distinct closed characteristics on \Sigma is finite, there exists at least an elliptic one among them, and there exist at least \varrho_n(\Sigma)-1 of them possessing irrational mean indices. If this total number is at most 2\varrho_n(\Sigma) -2, there exist at least two elliptic ones among them.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/1327601026a47c770f580a665e5cef67b01aadbf

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In Applied Mathematics

1. Chun-gen Liu
2. Y. Long
3. Wei Wang
4. Huagui Duan
5. Xijun Hu
6. Shanzhong Sun
7. Basak Z. Gurel
8. V. Ginzburg

In Geometry and Topology

1. Chun-gen Liu
2. Y. Long
3. Wei Wang
4. Xijun Hu
5. Huagui Duan
6. Shanzhong Sun
7. Duanzhi Zhang

Citations in the corpus

• The Relation of the Morse Index of Closed Geodesics with the Maslov–type Index of Symplectic Paths (2005, background): “[ 18 ]. From Theorem 1.1, we note that all these iteration index formulae that appear in these papers mentioned, are valid for the Morse indices (ind(c m ), null(c m )). So we can get a good”
• Seifert Conjecture in the Even Convex Case (2013, unknown): “per, we write (iL0(γ,k),νL0(γ,k)) = (iL0(γk),νL 0(γ k)) for any symplectic path γ∈ Pτ(2n) and k∈ N, where γk is defined by Definition 2.5. We have Lemma 5.1. (Theorem 1.5 and of [19] and Theorem 4.3 of [25]) Let γj ∈ Pτ j(2n) for j= 1,···,q. Let Mj = γ2 j (2τj) = Nγj(τj)−1Nγj(τj), for j= 1,···,q. Supp”
• Index and Stability of Symmetric Periodic Orbits in Hamiltonian Systems with Application to Figure-Eight Orbit (2009, background): “Recently, Long and Zhu [ LZ1 ] prove that there is at least one elliptic closed characteristic if the number is finite on any compact convex energy hypersurface in R 2n .”
• The enhanced common index jump theorem for symplectic paths and non-hyperbolic closed geodesics on Finsler manifolds (2015, methodology): “ersity,Beijing100871 ThePeople’sRepublicofChina September23,2015 Abstract In this paper, we first generalize the common index jump theorem for symplectic matrix paths proved in 2002 by Long and Zhu in [LoZ], and get an enhanced version of it. As its applications, we further prove that for a compact s”
• Iterated index formulae for closed geodesics with applications (2002, methodology): “[ 9 ], we have the following results in Morse indices whose proofs follow directly from those in ref. [ 9 ], Theorem 4.11 of ref.”
• Multiplicity of closed characteristics on symmetric convex hypersurfaces in $\mathbb{R}^{2n}$ (2001, background): “1 of [20]) Suppose #J̃ (Σ) < +∞, there exist an injection map p = p(Σ, α):N → J̃ (Σ, α) × N such that for any k ∈ N, (τ, x) ∈ J (Σ, α) and m ∈ N satisfying p(k) = ([(τ, x)],m), there hold f ′ α(u x m) = 0 and fα(u x m) = ck, (32) imτ (x ) ≤ 2k − 2 + n ≤ imτ (x) + νmτ (x)− 1, (33)”