Maurice de Gosson, Serge de Gosson, Paolo Piccione
Using invariance by fixed-endpoints homotopies and a generalized notion of symplectic Cayley transform, we prove a product formula for the Conley--Zehnder index of continuous paths with arbitrary endpoints in the symplectic group. We discuss two applications of the formula, to the metaplectic group and to periodic solutions of Hamiltonian systems.
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We calculate the character of the Weil representation using previous results which express the Weyl symbol of metaplectic operators in terms of the symplectic Cayley transform and the Conley--Zehnder index.
We prove a useful relation between the Conley-Zehnder indices of the Reeb vector flow action along periodic orbits in prequantization bundles and the orbifold Chern class of the base symplectic orbifolds motivated by the well-known case of manifolds. We also apply this method to primary examples.
We determine the Conley-Zehnder indices of all periodic orbits of the rotating Kepler problem for energies below the critical Jacobi energy. Consequently, we show the universal cover of the bounded component of the regularized energy hypersurface is dynamically convex. Moreover, in the universal cover there is always precisely one periodic orbit with Conley-Zehnder index 3, namely the lift of the doubly covered retrograde circular orbit.
We investigate the probability distribution of Conley-Zehnder indices associated with Brownian random paths on Sp(2n, R) that start at the identity. In the case of n = 1, we prove that the distribution has the same moment asymptotics as the standard random walk on the real line. We also present numerical evidence suggesting that the same asymptotics should hold for general n.
The Conley-Zehnder index associates an integer to any continuous path of symplectic matrices starting from the identity and ending at a matrix which does not admit 1 as an eigenvalue. We give new ways to compute this index. Robbin and Salamon define a generalization of the Conley-Zehnder index for any continuous path of symplectic matrices; this generalization is half integer valued. It is based on a Maslov-type index that they define for a continuous path of Lagrangians in a symplectic vector space $(W,\barΩ)$, having chosen a given reference Lagrangian $V$. Paths of symplectic endomorphisms …
Main results are, firstly, a generalization of the Conley-Zehnder index from ODEs to the delay equation at hand and, secondly, the equality of the Morse index and the clockwise normalized Conley-Zehnder index.
We give here a self contained and elementary introduction to the Conley-Zehnder index for a path of symplectic matrices. We start from the definition of the index as the degree of a map into the circle for a path starting at the identity and ending at a matrix for which 1 is not an eigenvalue. We prove some properties which characterize this index using normal forms for symplectic matrices obtained from geometrical considerations. We explore the relations to Robbin-Salamon index for paths of Lagrangians. We give an axiomatic characterization of the generalization of the Conley-Zehnder index fo…
The random Arnold Conjecture: a new probabilistic Conley-Zehnder Theory for symplectic maps
本文首次在概率框架下发展 Arnold 猜想的 Conley-Zehnder 理论，证明了在类似经典条件成立时，维度大于2的准周期辛扭转映射几乎必然具有无穷多个不动点，并提供了相关概率定理。
The lunar periods of our moon -- the companion of the Earth -- which date back to the Babylonians until around 600 BCE, are 29.53 days for the synodic, 27.55 days for the anomalistic and 27.21 days for the draconitic month. In this paper we define and compute these periods in terms of Floquet multipliers and Conley--Zehnder indices for planar periodic orbits in the spatial Hill lunar problem, which is a limit case of the spatial circular restricted three body problem. For very low energies, we are able to prove analytically the existence of the families of planar direct (family $g$) and retrog…
Conley-Zehnder Indices of Spatial Rotating Kepler Problem
本文通过辛拓扑方法研究空间旋转开普勒问题中的周期轨道，利用角动量和拉普拉斯-Runge-李特尔向量进行分类，并计算相关指数以贡献辛同调。
We prove the Conley conjecture for negative monotone, closed symplectic manifolds, i.e., the existence of infinitely many periodic orbits for Hamiltonian diffeomorphisms of such manifolds.
The Conley-Zehnder index of a minimal orbit and existence of a positive hyperbolic orbit
本文针对非退化接触三维球面的正双曲轨道存在性问题，在极小周期轨道的Conley-Zehnder指标≥3的条件下证明了简单正双曲轨道存在，并推出动力凸情形及四维欧氏空间中通有严格凸能量超曲面的相关结论。
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…
On the generic Conley conjecture
本文针对闭辛流形上哈密顿微分同胚的一般性Conley猜想，证明了在非常广泛的辛流形类中，一般哈密顿微分同胚具有无穷多个简单可缩周期轨道。
We prove a generalization of the Conley conjecture: Every Hamiltonian diffeomorphism of a closed symplectic manifold has infinitely many periodic orbits if the first Chern class vanishes over the second fundamental group. In particular, we this removes the rationality condition from similar results. The proof in the irrational case involves several new ideas including the definition and the properties of the filtered Floer homology for Hamiltonians on irrational manifolds. We also develop a method of localizing the filtered Floer homology for short action intervals using a direct sum decomposi…
In this paper we connect algebraic properties of the pair-of-pants product in local Floer homology and Hamiltonian dynamics. We show that for an isolated periodic orbit the product is non-uniformly nilpotent and use this fact to give a simple proof of the Conley conjecture for closed manifolds with aspherical symplectic form. More precisely, we prove that on a closed symplectic manifold the mean action spectrum of a Hamiltonian diffeomorphism with isolated periodic orbits is infinite.
Non-convex Mather's theory and the Conley conjecture on the cotangent bundle of the torus
本文通过辛同调化方法证明了非凸马瑟理论中关于圆柱的康利猜想及周期轨道的存在性。
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 …
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 $1$.
We show that whenever a closed symplectic manifold admits a Hamiltonian diffeomorphism with finitely many simple periodic orbits, the manifold has a spherical homology class of degree two with positive symplectic area and positive integral of the first Chern class. This theorem encompasses all known cases of the Conley conjecture (symplectic CY and negative monotone manifolds) and also some new ones (e.g., weakly exact symplectic manifolds with non-vanishing first Chern class). The proof hinges on a general Lusternik-Schnirelmann type result that, under some natural additional conditions, the …
We study the twisted Weyl symbol of metaplectic operators; this requires the definition of an index for symplectic paths related to the Conley-Zehnder index. We thereafter define a metaplectically covariant algebra of pseudo-differential operators acting on functions on symplectic space.
We use the minimal coupling procedure of Sternberg and Weinstein and our pseudo-symplectic capacity theory to prove that every closed symplectic submanifold in any symplectic manifold has an open neighborhood with finite ($π_1$-sensitive) Hofer-Zehnder symplectic capacity. Consequently, the Weinstein conjecture holds near closed symplectic submanifolds in any symplectic manifold.
We study the Weyl representation of metaplectic operators associated to a symplectic matrix having no non-trivial fixed point, and justify a formula suggested in earlier work of Mehlig and Wilkinson. We give precise calculations of the associated Maslov-type indices; these indices intervene in a crucial way in Gutzwiller's formula of semiclassical mechanics, and are simply related to an index defined by Conley and Zehnder.
We introduce the concept of Hofer-Zehnder $G$-semicapacity (or $G$-sensitive Hofer-Zehnder capacity) and prove that given a geometrically bounded symplectic manifold $(M,ω)$ and an open subset $N \subset M$ endowed with a Hamiltonian free circle action $ϕ$ then $N$ has bounded Hofer-Zehnder $G_ϕ$-semicapacity, where $G_ϕ\subset π_1(N)$ is the subgroup generated by the homotopy class of the orbits of $ϕ$. In particular, $N$ has bounded Hofer-Zehnder capacity. We give two types of applications of the main result. Firstly, we prove that the cotangent bundle of a compact manifold endowed with a fr…
On the Hofer-Zehnder conjecture for semipositive symplectic manifolds
论文研究半正辛流形上霍弗-泽亨德猜想，证明若哈密顿同胚的可缩固定点数超过流形总贝蒂数，则必有无限多个周期点，推广了谢卢金的结果。
We introduce the notion of a symplectic capacity relative to a coisotropic submanifold of a symplectic manifold, and we construct two examples of such capacities through modifications of the Hofer-Zehnder capacity. As a consequence, we obtain a non-squeezing theorem for symplectic embeddings relative to coisotropic constraints and existence results for leafwise chords on energy surfaces.
Generalizations of Ekeland-Hofer and Hofer-Zehnder symplectic capacities and applications
本文基于受Clarke和Ekeland工作启发的Hamilton边值问题，构造了Ekeland-Hofer和Hofer-Zehnder辛容量的类似物，并推广了Weinstein猜想、表示公式及Neduv定理等重要结果。
We use Floer homology to study the Hofer-Zehnder capacity of neighborhoods near a closed symplectic submanifold M of a geometrically bounded and symplectically aspherical ambient manifold. We prove that, when the unit normal bundle of M is homologically trivial in degree dim(M) (for example, if codim(M) > dim(M)), a refined version of the Hofer-Zehnder capacity is finite for all open sets close enough to M. We compute this capacity for certain tubular neighborhoods of M by using a squeezing argument in which the algebraic framework of Floer theory is used to detect nontrivial periodic orbits. …
The sharp energy-capacity inequality on convex symplectic manifolds
本文将Usher关于闭辛流形的尖锐能量-容量不等式推广到凸辛流形，建立了Hofer-Zehnder容量与位移能量之间的不等式关系。
We study the relationship between a homological capacity $c_{\mathrm{SH}^+}(W)$ for Liouville domains $W$ defined using positive symplectic homology and the existence of periodic orbits for Hamiltonian systems on $W$: If the positive symplectic homology of $W$ is non-zero, then the capacity yields a finite upper bound to the $π_1$-sensitive Hofer-Zehnder capacity of $W$ relative to its skeleton and a certain class of Hamiltonian diffeomorphisms of $W$ has infinitely many non-trivial contractible periodic points. En passant, we give an upper bound for the spectral capacity of $W$ in terms of th…
We exhibit many examples of closed symplectic manifolds on which there is an autonomous Hamiltonian whose associated flow has no nonconstant periodic orbits (the only previous explicit example in the literature was the torus T^2n (n\geq 2) with an irrational symplectic structure). The underlying smooth manifolds of our examples include, for instance: the K3 surface and also infinitely many smooth manifolds homeomorphic but not diffeomorphic to it; infinitely many minimal four-manifolds having any given finitely-presented group as their fundamental group; and simply connected minimal four-manif…
We introduce the concept of pseudo symplectic capacities which is a mild generalization of that of symplectic capacities. As a generalization of the Hofer-Zehnder capacity we construct a Hofer-Zehnder type pseudo symplectic capacity and estimate it in terms of Gromov-Witten invariants. The (pseudo) symplectic capacities of Grassmannians and some product symplectic manifolds are computed. As applications we first derive some general nonsqueezing theorems that generalize and unite many previous versions, then prove the Weinstein conjecture for cotangent bundles over a large class of symplectic u…
A Lower Bound of the Hofer-Zehnder Capacity via Delzant Polytopes
本文利用辛环面流形的哈密顿子圆环作用周期与 Delzant 多胞体的关系，发展出 Delzant 多胞体的一个不变量，从而给出了 Hofer-Zehnder 容量的下界。
On the Hofer-Zehnder conjecture for non-contractible periodic orbits in Hamiltonian dynamics
本文研究哈密顿同胚在闭合辛流形上的周期轨道数量问题，证明非收缩周期轨道的霍弗-泽亨德猜想在广泛辛流形类别中成立。
Gutzwiller's famous semiclassical trace formula plays an important role in theoretical and experimental quantum mechanics with tremendous success. We review the physical derivation of this deep periodic orbit theory in terms of the phase space formulation with an view towards the Hamiltonian dynamical systems. The Maslov phase appearing in the trace formula is clarified by Meinrenken as Conley-Zhender index for periodic orbits of Hamiltonian systems. We also survey and compare various versions of Maslov indices to establish this fact. A refinement and improvement to Conley-Zehnder's index theo…
Inspired by the work of G. Lu on pseudo symplectic capacities we obtain several results on the Gromov width and the Hofer--Zehnder capacity of Hermitian symmetric spaces of compact type. Our results and proofs extend those obtained by Lu for complex Grassmannians to Hermitian symmetric spaces of compact type. We also compute the Gromov width and the Hofer--Zehnder capacity for Cartan domains and their products.
We find bounds for the Hofer-Zehnder capacity of coadjoint orbits of compact Lie groups with respect to the Kostant--Kirillov--Souriau symplectic form in terms of the combinatorics of their Bruhat graph. We show that our bounds are sharp for coadjoint orbits of the unitary group and equal to the diameter of a weighted Cayley graph.
The main theme of this paper is a relative version of the almost existence theorem for periodic orbits of autonomous Hamiltonian systems. We show that almost all low levels of a function on a geometrically bounded symplectically aspherical manifold carry contractible periodic orbits of the Hamiltonian flow, provided that the function attains its minimum along a closed symplectic submanifold. As an immediate consequence, we obtain the existence of contractible periodic orbits on almost all low energy levels for twisted geodesic flows with symplectic magnetic field. We give examples of functions…
Franks' dichotomy for toric manifolds, Hofer-Zehnder conjecture, and gauged linear sigma model
本文利用规范线性sigma模型及其整体形变研究辛商空间的哈密顿动力学，证明了紧致环面辛流形上固定点同调计数超过总Betti数的哈密顿微分同胚必有无穷多个简单周期点，从而推广了Franks二分性并证实了Hofer-Zehnder猜想。
Let $(M,ω)$ be an aspherical symplectic manifold, which is closed or convex. Let $U$ be an open set in $M$, which admits a circle action generated by an autonomous Hamiltonian $H \in C^\infty(U)$, such that each orbit of the circle action is not contractible in $M$. Under these assumptions, we prove that the Hofer-Zehnder capacity of $U$ is bounded by the Hofer norm of $H$. The proof uses a variant of the energy-capacity inequality, which is proved by the theory of action selectors.
Remarks on periodic points, flux, and the mean index in symplectic dynamics
本文证明闭辛Calabi-Yau流形上非退化端点辛同伦在通量相关同调非消失条件下具有任意大周期简单周期点，并应用于标准辛4-环面。
In this paper we prove that the symplectic Brunn-Minkowski inequality, established by Artstein-Avidan and Ostrover for the Ekeland-Hofer-Zehnder capacity, fails to hold for the Gutt-Hutchings capacities of index greater than one.
We use the criteria of Lalonde and McDuff to determine a new class of examples of length minimizing paths in the group $Ham(M)$. For a compact symplectic manifold $M$ of dimension two or four, we show that a path in $Ham(M)$, generated by an autonomous Hamiltonian and starting at the identity, which induces no non-constant closed trajectories of points in $M$, is length minimizing among homotopic paths. The major step in the proof involves determining an upper bound for the Hofer-Zehnder capacity for symplectic manifolds of the type $(M \times D(a))$ where $M$ is compact and has dimension two …
On the symplectic size of convex polytopes
本文引入了一个计算$\mathbb{R}^{2n}$中凸多胞体Ekeland-Hofer-Zehnder容量的组合公式，并证明了该容量的次可加性。
We improve the estimates for the Ekeland--Hofer--Zehnder capacity of convex bodies by Gluskin and Ostrover. In the course of our argument we show that a closed characteristic of minimal action on the boundary of a centrally symmetric convex body in $\mathbb R^{2n}$ must itself be centrally symmetric, with generalizations to some other types of symmetry.
On the Maslov-type index for general paths of symplectic matrices
本文为具有任意端点的一般辛路径直接构造了一种Maslov型指标，该指标部分推广了Conley-Zehnder-Long指标，且构造不依赖起点是否为单位矩阵，并可与Cappell-Lee-Miller指标建立联系。
In this note we link symplectic and convex geometry by relating two seemingly different open conjectures: a symplectic isoperimetric-type inequality for convex domains, and Mahler's conjecture on the volume product of centrally symmetric convex bodies. More precisely, we show that if for convex bodies of fixed volume in the classical phase space the Hofer-Zehnder capacity is maximized by the Euclidean ball, then a hypercube is a minimizer for the volume product among centrally symmetric convex bodies.
We show that the actions and indexes of fixed points of a Hamiltonian diffeomorphism with finitely many periodic points must satisfy certain relations, provided that the quantum cohomology of the ambient manifold meets an algebraic requirement satisfied for projective spaces, Grassmannians and many other manifolds. We also refine a previous result on the Conley conjecture for negative monotone symplectic manifolds, due to the second and third authors, and show that a Hamiltonian diffeomorphism of such a manifold must have simple periodic orbits of arbitrarily large period whenever its fixed po…
In this paper we use the Ekeland-Hofer-Zehnder symplectic capacity to provide several bounds and inequalities for the length of the shortest periodic billiard trajectory in a smooth convex body in ${\mathbb R}^{n}$. Our results hold both for classical billiards, as well as for the more general case of Minkowski billiards.
Let M be a weakly monotone symplectic manifold, and H be a time-dependent Hamiltonian; we assume that the periodic orbits of the corresponding time-dependent Hamiltonian vector field are non-degenerate. We construct a refined version of the Floer chain complex associated to these data and any regular covering of M, and derive from it new lower bounds for the number of periodic orbits. We prove in particular that if the fundamental group of M is finite and solvable or simple, then the number of periodic orbits is not less than the minimal number of generators of the fundamental group. For a gen…
Continuing the study of Hamiltonian pseudo-rotations of projective spaces, we focus on the conjecture that the fixed-point data set (the actions and the linearized flows at one-periodic orbits) of a pseudo-rotation exactly matches that data for a suitable unique true rotation even though the two maps can have very different dynamics. We prove this conjecture in several instances, e.g., for strongly non-degenerate pseudo-rotations of ${{\mathbb C}{\mathbb P}}^2$ with some notable exceptions, which we call ghost pseudo-rotations. The existence of ghost pseudo-rotations is a completely open quest…
Let $(M,ω)$ be a symplectic manifold, $N\subseteq M$ a coisotropic submanifold, and $Σ$ a compact oriented (real) surface. I define a natural Maslov index for each continuous map $u:Σ\to M$ that sends every connected component of $\partialΣ$ to some isotropic leaf of $N$. This index is real valued and generalizes the usual Lagrangian Maslov index. The idea is to use the linear holonomy of the isotropic foliation of $N$ to compensate for the loss of boundary data in the case codimension $N<\dim M/2$. The definition is based on the Salamon-Zehnder (mean) Maslov index of a path of linear symplect…
Computing the EHZ capacity is NP-hard
本文通过将二部竞赛的反馈弧集问题归约为单纯形 EHZ 容量计算，证明计算多面体的 Ekeland-Hofer-Zehnder 容量具有 NP 难复杂性。
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