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

Multiplicity and ellipticity of closed characteristics on compact star-shaped hypersurfaces in ${\bf R}^{2n}$

In this paper, we firstly generalize some theories developed by I. Ekeland and H. Hofer in [EkH] for closed characteristics on compact convex hypersurfaces in ${\bf R}^{2n}$ to star-shaped hypersurfaces. As applications, we use Ekeland-Hofer theory and index iteration theory to prove that if a compact star-shaped hypersuface in ${\bf R}^4$ satisfying some suitable pinching condition carries exactly two geometrically distinct closed characteristics, then both of them must be elliptic. We also conclude that the theory developed by Y. Long and C. Zhu in [LoZ] still holds for dynamically convex st…

Remarks on the systoles of symmetric convex hypersurfaces and symplectic capacities

In this note we study the systoles of convex hypersurfaces in $\mathbb{R}^{2n}$ invariant under an anti-symplectic involution. We investigate a uniform upper bound of the ratio between the systole and the symmetric systole of the hypersurfaces using symplectic capacities from Floer theory. We discuss various concrete examples in which the ratio can be understood explicitly.

接触流形中的鲁棒非凸超曲面

Robustly non-convex hypersurfaces in contact manifolds

本文通过首次在接触情形构造动力学混合器，给出5维及以上接触流形中不能被凸超曲面$C^2$逼近的鲁棒非凸超曲面首批示例。

凸超曲面与鲁棒异维动力学

Convex hypersurfaces and robust heterodimensional dynamics

研究五维及以上接触流形中闭可定向超曲面，通过结合异宿轨道的凸性障碍和鲁棒异维动力学进展，证明可经任意\(C^0\)小同痕与鲁棒非凸超曲面同痕，加强了相关结果。

Multiplicity and stability of closed characteristics on compact convex P-cyclic symmetric hypersurfaces in ${\bf R}^{2n}$

Let $Σ$ be a compact convex hypersurface in ${\bf R}^{2n}$ which is P-cyclic symmetric, i.e., $x\in Σ$ implies $Px\inΣ$ with P being a $2n\times2n$ symplectic orthogonal matrix and satisfying $P^k=I_{2n}$, $ker(P^l-I_{2n})=0$ for $1\leq l< k$, where $n, k\geq2$. In this paper, we prove that there exist at least $n$ geometrically distinct closed characteristics on $Σ$, which solves a longstanding conjecture about the multiplicity of closed characteristics for a broad class of compact convex hypersurfaces with symmetries(cf.,Page 235 of \cite{Eke1}). Based on the proof, we further prove that if …

Stability of closed characteristics on compact convex hypersurfaces in R^{2n}

Let $Σ\subset \R^{2n}$ with $n\geq2$ be any $C^2$ compact convex hypersurface and only has finitely geometrically distinct closed characteristics. Based on Y.Long and C.Zhu 's index jump methods \cite{LoZ1}, we prove that there are at least two geometrically distinct elliptic closed characteristics, and moreover, there exist at least $\varrho_{n} (Σ)$ ($\varrho_{n}(Σ)\geq[\frac{n}{2}]+1$) geometrically distinct closed characteristics such that for any two elements among them, the ratio of their mean indices is irrational number.