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

Xiaorui Li; Zhenxiong Li; Hui Liu

doi:10.3934/dcds.2024067

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2024, Volume 44, Issue 11: 3491-3506. Doi: 10.3934/dcds.2024067

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

School of Mathematics and Statistics, Wuhan University, China

^*Corresponding author: Zhenxiong Li
^*Corresponding author: Zhenxiong Li

Received: November 2023

Revised: April 2024

Early access: May 2024

Published: November 2024

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Abstract

In this paper, let $ P = \begin{pmatrix}\cos\frac{2\pi}{p} &-\sin\frac{2\pi}{p}\\ \sin\frac{2\pi}{p} &\cos\frac{2\pi}{p} \end{pmatrix}^{\diamond n} $ be a rotational matrix, and $ \Sigma \subset\mathbb{R}^{2n} $ be a $ P $-symmetric compact convex hypersurface which is $ (r,R) $-pinched with $ \frac{R}{r}<\sqrt{\frac{2p+1}{p+1}} $, where the integer $ p\geq 2 $. Then $ \Sigma $ carries at least two elliptic $ P $-cyclic closed characteristics; moreover, $ \Sigma $ carries at least $ \lceil\frac{(n-1)p}{2(p+1)}\rceil+\lceil \frac{n-1}{2} \rceil $ non-hyperbolic $ P $-cyclic closed characteristics. The proofs are based on the Maslov $ P $-index iteration theory and comparison theorems on indices.

Keywords:

P-symmetric compact convex hypersurfaces,
P-cyclic closed characteristics,
Hamiltonian systems,
Maslov P-index iteration,
stability.

Mathematics Subject Classification: Primary: 37J12, 37J25; Secondary: 53D12.

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