Brake orbits on compact symmetric dynamically convex reversible hypersurfaces on $ \mathbb{R}^\text{2n} $

Zhongjie Liu; Duanzhi Zhang

doi:10.3934/dcds.2019169

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2019, Volume 39, Issue 7: 4187-4206. Doi: 10.3934/dcds.2019169

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Brake orbits on compact symmetric dynamically convex reversible hypersurfaces on $ \mathbb{R}^\text{2n} $

Zhongjie Liu and
Duanzhi Zhang^,

School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China

Author Bio: E-mail address: lzj0207@yeah.net (Z. Liu); E-mail address: zhangdz@nankai.edu.cn
^* Corresponding author. Partially supported by the NSF of China (17190271, 11422103, 11771341) and Nankai University. E-mail: zhangdz@nankai.edu.cn
^* Corresponding author. Partially supported by the NSF of China (17190271, 11422103, 11771341) and Nankai University. E-mail: zhangdz@nankai.edu.cn

Received: September 30, 2018

Revised: November 30, 2018

Published: April 2019

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Abstract

In this paper, we consider multiplicity of brake orbits on compact symmetric dynamically convex reversible hypersurfaces in $\mathbb{R}^{2n}$. We prove that there exist at least $[\frac{n+1}{2}]$ geometrically distinct closed characteristics on dynamically convex hypersurface $\Sigma$ in $R^{2n}$ with the symmetric and reversible conditions, i.e. $\Sigma = -\Sigma$ and $N\Sigma = \Sigma$, where $N = diag(-I_{n}, I_{n})$. For $n\geq2$, we prove that there are at least 2 symmetric brake orbits on $\Sigma$, which generalizes Kang's result in [8] of 2013.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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