Multiplicity of closed characteristics on $ P $-symmetric compact convex hypersurfaces in $ \mathbb{R}^{2n} $

Lei Liu; Li Wu

doi:10.3934/dcds.2020378

Article Contents

2021, Volume 41, Issue 6: 2635-3652. Doi: 10.3934/dcds.2020378

This issue Previous Article Second order estimates for complex Hessian equations on Hermitian manifolds Next Article Normalized solutions for nonlinear coupled fractional systems: Low and high perturbations in the attractive case

Multiplicity of closed characteristics on $ P $-symmetric compact convex hypersurfaces in $ \mathbb{R}^{2n} $

Lei Liu^1, , and
Li Wu^1,

1.
School of Mathematics Department, Shandong University, Jinan 250100, China

^* Corresponding author: Lei Liu
^* Corresponding author: Lei Liu

Received: May 2020

Revised: September 2020

Early access: November 12, 2020

Published online: November 12, 2020

The first author is supported by NSFC grant No.11425105.

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Abstract

There is a long standing conjecture that there are at least $ n $ closed characteristics on any compact convex hypersurface $ \Sigma $ in $ \mathbb{R}^{2n} $. In this paper, we provide some new estimates and prove that there are at least $ [\frac{3n}{4}] $ closed characteristics on $ \Sigma $ for any positive integer $ n $, where $ \Sigma $ satisfies $ \Sigma = P\Sigma $ for a certain class of symplectic matrix $ P $. These results are not considered in previous papers.

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Mathematics Subject Classification: Primary: 58E05, 70H12, 34C25.

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