scholarly journals $P$-cyclic symmetric closed characteristics on compact convex $P$-cyclic symmetric hypersurface in R2n

2013 ◽  
Vol 33 (2) ◽  
pp. 947-964 ◽  
Author(s):  
Duanzhi Zhang ◽  
Keyword(s):  
Compact Convex ◽  
Cyclic Symmetric ◽  
Closed Characteristics ◽  
Symmetric Closed Characteristics

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}}.

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