Keplerian orbits through the Conley-Zehnder index

Length minimizing property, Conley-Zehnder index and $C^1$-perturbations of Hamiltonian functions

The main purpose of this paper is to study the length minimizing property of Hamiltonian paths on closed symplectic manifolds $(M,ω)$ such that there are no spherical homology class $A \in H_2(M)$ with $$ ω(A) > 0 \quad \text{and} \quad -n \leq c_1(A) < 0, $$ which we call {\it very strongly semi-positive}. We introduce the notion of {\it positively $μ$-undertwisted} Hamiltonian paths and prove that any positively undertwisted quasi-autonomous Hamiltonian path is length minimizing in its homotopy class as long as it has a fixed maximum and a fixed minimum point that are generically under-twist…

The $E$-cohomological Conley Index, Cup-Lengths and the Arnold Conjecture on $T^{2n}$

We give a new proof of the strong Arnold conjecture for $1$-periodic solutions of Hamiltonian systems on tori, that was first shown by C. Conley and E. Zehnder in 1983. Our proof uses other methods and is shorter than the previous one. We first show that the $E$-cohomological Conley index, that was introduced by the first author recently, has a natural module structure. This yields a new cup-length and a lower bound for the number of critical points of functionals. Then an existence result for the $E$-cohomological Conley index, which applies to the setting of the Arnold conjecture, paves the …

Computing Reeb dynamics on 4d convex polytopes

We study the combinatorial Reeb flow on the boundary of a four-dimensional convex polytope. We establish a correspondence between "combinatorial Reeb orbits" for a polytope, and ordinary Reeb orbits for a smoothing of the polytope, respecting action and Conley-Zehnder index. One can then use a computer to find all combinatorial Reeb orbits up to a given action and Conley-Zehnder index. We present some results of experiments testing Viterbo's conjecture and related conjectures. In particular, we have found some new examples of polytopes with systolic ratio $1$.

On the Maslov class rigidity for coisotropic submanifolds

We define the Maslov index of a loop tangent to the characteristic foliation of a coisotropic submanifold as the mean Conley--Zehnder index of a path in the group of linear symplectic transformations, incorporating the "rotation" of the tangent space of the leaf -- this is the standard Lagrangian counterpart -- and the holonomy of the characteristic foliation. Furthermore, we show that, with this definition, the Maslov class rigidity extends to the class of the so-called stable coisotropic submanifolds including Lagrangian tori and stable hypersurfaces.

紧$3$-球面上弱凸Reeb流的亏格为零的横截叶状结构

Genus zero transverse foliations for weakly convex Reeb flows on the tight $3$-sphere

本文研究紧3-球面上带指定有限个双曲且互不环绕的2指标Reeb轨道的弱凸Reeb流，给出其构成亏格零横截叶状结构绑定轨道的条件，并在实解析情形下证明稳定/不稳定流形分支不重合时Reeb流拓扑熵为正。

类彗星型周期运动及其平面外分支在地球-月球CR3BP中的研究：一种计算辛分析

Comet-type periodic motions and their out-of-plane bifurcations in the Earth-Moon CR3BP: a computational symplectic analysis

本文通过辛不变量分析，研究了地球-月球CR3BP中类彗星型周期运动的生成、稳定性及高阶自共振分支特性，揭示了其轨道特征与共振关系。