2018/02/14 by Lei Liu, Liu, Lei, Li Wu +1
Mathematics · #34C25 #58E05 #70H12 #Advanced Algebra and Geometry #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1802.04935
openalex publication_date 2018/02/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
There is a long standing conjecture that there are at least n closed characteristics for any compact convex hypersurface Σ in ℝ2n, and the symmetric case, i.e. Σ=-Σ, has already been proved by C. Liu, Y. Long and C. Zhu in [Math. Ann., 323(2002), pp. 201-215]. In this paper, we extend the result in that paper to the P-symmetric case Σ=PΣ for a certain class of symplectic matrix P, and prove that there are at least [(3n)/(4)] closed characteristics on Σ for any positive integer n, where [a]:=sup\l∈ℤ,l≤ a\. To obtain our result, the key problem is to estimate (3.13) in which the method is based on the theorem called Common Index Jump Theorem. By using the Bott-type iteration formulas of Maslov index and Maslov-type index for a certain kind of iteration symplectic path, we provide the some new estimations (4.9-4.11), which are not considered in other papers.