scholarly journals Resonance identities and stability of symmetric closed characteristics on symmetric compact star-shaped hypersurfaces

2015 ◽  
Vol 54 (4) ◽  
pp. 3753-3787 ◽  
Author(s):  
Hui Liu ◽  
Yiming Long
Keyword(s):  
Compact Star ◽  
Closed Characteristics ◽  
Symmetric Closed Characteristics

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>

scholarly journals Higher P-symmetric Ekeland-Hofer capacities

2012 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Kun Shi ◽  
Guangcun Lu
Keyword(s):  
Convex Bodies ◽  
Symplectic Space ◽  
Symmetric Convex ◽  
The Real ◽  
Closed Characteristics ◽  
The Relationship ◽  
Symmetric Closed Characteristics

<p style='text-indent:20px;'>This paper is devoted to the construction of analogues of higher Ekeland-Hofer symplectic capacities for <inline-formula><tex-math id="M2">\begin{document}$ P $\end{document}</tex-math></inline-formula>-symmetric subsets in the standard symplectic space <inline-formula><tex-math id="M3">\begin{document}$ (\mathbb{R}^{2n},\omega_0) $\end{document}</tex-math></inline-formula>, which is motivated by Long and Dong's study about <inline-formula><tex-math id="M4">\begin{document}$ P $\end{document}</tex-math></inline-formula>-symmetric closed characteristics on <inline-formula><tex-math id="M5">\begin{document}$ P $\end{document}</tex-math></inline-formula>-symmetric convex bodies. We study the relationship between these capacities and other capacities, and give some computation examples. Moreover, we also define higher real symmetric Ekeland-Hofer capacities as a complement of Jin and the second named author's recent study of the real symmetric analogue about the first Ekeland-Hofer capacity.</p>

Sign in / Sign up

Export Citation Format

Share Document