scholarly journals Multiplicity of closed characteristics on symmetric convex hypersurfaces in $\mathbb{R}^{2n}$

2002 ◽  
Vol 323 (2) ◽  
pp. 201-215 ◽  
Author(s):  
Chun-gen Liu ◽  
Yiming Long ◽  
Chaofeng Zhu
Keyword(s):  
Symmetric Convex ◽  
Closed Characteristics ◽  
Convex Hypersurfaces

Non-hyperbolic P-Invariant Closed Characteristics on Partially Symmetric Compact Convex Hypersurfaces

2018 ◽  
Vol 18 (4) ◽  
pp. 763-774
Author(s):  
Hui Liu ◽  
Gaosheng Zhu
Keyword(s):  
Compact Convex ◽  
Convex Hypersurface ◽  
Closed Characteristics ◽  
Compact Convex Hypersurfaces ◽  
Partially Symmetric ◽  
Convex Hypersurfaces

AbstractLet {n\geq 2} be an integer, {P=\mathrm{diag}(-I_{n-\kappa},I_{\kappa},-I_{n-\kappa},I_{\kappa})} for some integer {\kappa\in[0,n]}, and let {\Sigma\subset{\mathbb{R}}^{2n}} be a partially symmetric compact convex hypersurface, i.e., {x\in\Sigma} implies {Px\in\Sigma}, and {(r,R)}-pinched. In this paper, we prove that when {{R/r}<\sqrt{5/3}} and {0\leq\kappa\leq[\frac{n-1}{2}]}, there exist at least {E(\frac{n-2\kappa-1}{2})+E(\frac{n-2\kappa-1}{3})} non-hyperbolic P-invariant closed characteristics on Σ. In addition, when {{R/r}<\sqrt{3/2}}, {[\frac{n+1}{2}]\leq\kappa\leq n} and Σ carries exactly nP-invariant closed characteristics, then there exist at least {2E(\frac{2\kappa-n-1}{4})+E(\frac{n-\kappa-1}{3})} non-hyperbolic P-invariant closed characteristics on Σ, where the function {E(a)} is defined as {E(a)=\min{\{k\in{\mathbb{Z}}\mid k\geq a\}}} for any {a\in\mathbb{R}}.

scholarly journals Multiplicity of closed Reeb orbits on dynamically convex $ \mathbb{R}P^{2n-1} $ for $ n\geq2 $

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Hui Liu ◽  
Ling Zhang
Keyword(s):  
Compact Convex ◽  
Compact Star ◽  
Main Ingredient ◽  
Closed Characteristics ◽  
Compact Convex Hypersurfaces ◽  
Convex Hypersurfaces

<p style='text-indent:20px;'>In this paper, we prove that there exist at least two non-contractible closed Reeb orbits on every dynamically convex <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{R}P^{2n-1} $\end{document}</tex-math></inline-formula>, and if all the closed Reeb orbits are non-degenerate, then there are at least <inline-formula><tex-math id="M4">\begin{document}$ n $\end{document}</tex-math></inline-formula> closed Reeb orbits, where <inline-formula><tex-math id="M5">\begin{document}$ n\geq2 $\end{document}</tex-math></inline-formula>, the main ingredient is that we generalize some theories developed by I. Ekeland and H. Hofer for closed characteristics on compact convex hypersurfaces in <inline-formula><tex-math id="M6">\begin{document}$ {{\bf R}}^{2n} $\end{document}</tex-math></inline-formula> to symmetric compact star-shaped hypersurfaces. In addition, we use Ekeland-Hofer theory to give a new proof of a theorem recently by M. Abreu and L. Macarini that every dynamically convex symmetric compact star-shaped hypersurface carries an elliptic symmetric closed characteristic.</p>

Non-hyperbolic Closed Characteristics on Symmetric Compact Convex Hypersurfaces in R2n

2014 ◽  
Vol 14 (3) ◽  
Author(s):  
Hui Liu ◽  
Yiming Long ◽  
Wei Wang
Keyword(s):  
Compact Convex ◽  
Closed Characteristics ◽  
Compact Convex Hypersurfaces ◽  
Convex Hypersurfaces

AbstractIn this paper, let Σ ⊂ R

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