Umberto L. Hryniewicz, Pedro A. S. Salomão, Richard Siefring
We establish some new existence results for global surfaces of section of dynamically convex Reeb flows on the three-sphere. These sections often have genus, and are the result of a combination of pseudo-holomorphic curve methods with some elementary ergodic methods.
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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。
We characterize which closed Reeb orbits of a dynamically convex contact form on the 3-sphere bound disk-like global surfaces of section for the Reeb flow, without any genericity assumptions. We show that these global surfaces of section come in families, organized as open book decompositions. As an application we obtain new global surfaces of section for the Hamiltonian dynamics on strictly convex 3-dimensional energy levels.
We exhibit sufficient conditions for a finite collection of periodic orbits of a Reeb flow on a closed $3$-manifold to bound a positive global surface of section with genus zero. These conditions turn out to be $C^\infty$-generically necessary. Moreover, they involve linking assumptions on periodic orbits with Conley-Zehnder index ranging in a finite set determined by the ambient contact geometry. As an application, we reprove and generalize a classical result of Birkhoff on the existence of annulus-like global surfaces of section for geodesic flows on positively curved two-spheres.
Let M be a closed manifold whose based loop space is ``complicated''. Examples are rationally hyperbolic manifolds and manifolds whose fundamental group has exponential growth. We prove that the topological entropy of any Reeb flow on the spherization of T*M is positive. Moreover, given q in M, for almost every q' in M the number of Reeb orbits from the fiber over q to the fiber over q' grows exponentially in time. If M is a surface of higher genus, we also obtain exponential growth of the number of closed Reeb orbits.
The first result of this paper is that every contact form on $\mathbb{R} P^3$ sufficiently $C^\infty$-close to a dynamically convex contact form admits an elliptic-parabolic closed Reeb orbit which is $2$-unknotted, has self-linking number $-1/2$ and transverse rotation number in $(1/2,1]$. Our second result implies that any $p$-unknotted periodic orbit with self-linking number $-1/p$ of a dynamically convex Reeb flow on a lens space of order $p$ is the binding of a rational open book decomposition, whose pages are global surfaces of section. As an application we show that in the planar circul…
Closed Orbits of Dynamically Convex Reeb Flows: Towards the HZ- and Multiplicity Conjectures
本文证明星形域边界上的动态凸Reeb流至少有n个素闭轨道，改进了此前下界；在中心对称且非退化条件下，轨道数要么恰好n个要么无穷多，推广了Franks的2或无穷定理。
Global surfaces of section for dynamically convex Reeb flows on lens spaces
该研究证明了标准紧透镜空间上动态凸Reeb流存在特定闭轨道作为有理开书分解的绑定，还在Hénon-Heiles系统中验证了动态凸性，相关流可下降为另一透镜空间上的Reeb流。
In this paper we prove the existence of infinitely many closed Reeb orbits for a certain class of contact manifolds. This result can be viewed as a contact analogue of the Hamiltonian Conley conjecture. The manifolds for which the contact Conley conjecture is established are the pre-quantization circle bundles with aspherical base. As an application, we prove that for a surface of genus at least two with a non-vanishing magnetic field, the twisted geodesic flow has infinitely many periodic orbits on every low energy level.
Existence of a positive hyperbolic orbit in three-dimensional Reeb flows
本文证明在闭连通三维流形上，若Reeb流具有至少三条简单周期轨道，则必存在一条正双曲轨道，方法基于ECH谱不变量的Weyl定律和亏格零全纯曲线的紧致性。
On the knot types of periodic Reeb orbits of dynamically convex contact forms
研究揭示了标准接触3球面上无法由动态凸接触形式产生的横贯结类型，并指出这些结类型不出现于四维辛向量空间的严格凸能量水平的闭合特征中。
We use Lerman's contact cut construction to find a sufficient condition for Hamiltonian diffeomorphisms of compact surfaces to embed into a closed 3-manifold as Poincaré return maps on a global surface of section for a Reeb flow. In particular, we show that the irrational pseudorotations of the 2-disc constructed by Fayad-Katok embed into the Reeb flow of a dynamically convex contact form on the 3-sphere.
We classify global surfaces of section for the Reeb flow of the standard contact form on the 3-sphere, defining the Hopf fibration. As an application, we prove the degree-genus formula for complex projective curves, using an elementary degeneration process inspired by the language of holomorphic buildings in symplectic field theory.
Two irrationally elliptic closed orbits of Reeb flows on the boundary of star-shaped domain in $\mathbb{R}^{2n}$
本文证明在 $\mathbb{R}^{2n}$ 中星形域边界上具有恰好 $n$ 个素闭轨道的动态凸 Reeb 流中，至少两个轨道是无理椭圆，支持了相关猜想。
For $g>0$, we construct $g+1$ Legendrian embeddings of a surface of genus $g$ into $J^1(R^2)=R^5$ which lie in pairwise distinct Legendrian isotopy classes and which all have $g+1$ transverse Reeb chords ($g+1$ is the conjecturally minimal number of chords). Furthermore, for $g$ of the $g+1$ embeddings the Legendrian contact homology DGA does not admit any augmentation over $Z/2Z$, and hence cannot be linearized. We also investigate these surfaces from the point of view of the theory of generating families. Finally, we consider Legendrian spheres and planes in $J^1(S^2)$ from a similar perspec…
We exhibit the first examples of contact structures on $S^{2n-1}$ with $n\geq 4$ and on $S^3\times S^2$, all equipped with their standard smooth structures, for which every Reeb flow has positive topological entropy. As a new technical tool for the study of the volume growth of Reeb flows we introduce the notion of algebraic growth of wrapped Floer homology. Its power stems from its stability under several geometric operations on Liouville domains.
Elliptic bindings and the first ECH spectrum for convex Reeb flows on lens spaces
本文通过有理自连接数和 Conley-Zehnder 指标给出透镜空间中 Reeb 轨道为椭圆轨道的充分条件，并证明动态凸 L(3,1) 上的第一 ECH 谱等于特定圆盘型 Birkhoff 截面的接触面积下确界。
The main theme of this paper is the dynamics of Reeb flows with symmetries on the standard contact sphere. We introduce the notion of strong dynamical convexity for contact forms invariant under a group action, supporting the standard contact structure, and prove that in dimension $2n+1$ any such contact form satisfying a condition slightly weaker than strong dynamical convexity has at least $n+1$ simple closed Reeb orbits. For contact forms with antipodal symmetry, we prove that strong dynamical convexity is a consequence of ordinary convexity. In dimension five or greater, we construct examp…
Let $X \subset \mathbb{R}^4$ be a convex domain with smooth boundary $Y$. We use a relation between the extrinsic curvature of $Y$ and the Ruelle invariant $\text{Ru}(Y)$ of the natural Reeb flow on $Y$ to prove that there exist constants $C > c > 0$ independent of $Y$ such that \[c < \frac{\text{Ru}(Y)^2}{\text{vol}(X)} \cdot \text{sys}(Y) < C\] Here $\text{sys}(Y)$ is the systolic ratio, i.e. the square of the minimal period of a closed Reeb orbit of $Y$ divided by twice the volume of $X$. We then construct dynamically convex contact forms on $S^3$ that violate this bound using methods of Ab…
Torus knotted Reeb dynamics and the Calabi invariant
本文针对标准紧三维球面上允许标准横向正T(p,q)环面纽结为椭圆Reeb轨道的接触形式，建立次级Reeb轨道集的存在性，推导相关保面积微分同胚周期轨道平均作用量上界，发展纽结过滤嵌入接触同调并得到精确辛余维存在障碍。
We give a sharp lower bound for the number of geometrically distinct contractible periodic orbits of dynamically convex Reeb flows on prequantizations of symplectic manifolds that are not aspherical. Several consequences of this result are obtained, like a new proof that every bumpy Finsler metric on $S^n$ carries at least two prime closed geodesics, multiplicity of elliptic and non-hyperbolic periodic orbits for dynamically convex contact forms with finitely many geometrically distinct contractible closed orbits and precise estimates of the number of even periodic orbits of perfect contact fo…
We explore the relationship between contact forms on $\mathbb S^3$ defined by Finsler metrics on $\mathbb S^2$ and the theory developed by H. Hofer, K. Wysocki and E. Zehnder in \cite{HWZ,HWZ1}. We show that a Finsler metric on $\mathbb S^2$ with curvature $K\geq 1$ and with all geodesic loops of length $>π$ is dynamically convex and hence it has either two or infinitely many closed geodesics. We also explain how to explicitly construct $J$-holomorphic embeddings of cylinders asymptotic to Reeb orbits of contact structures arising from Finsler metrics on $\mathbb S^2$ with K=1 thus complementi…
Dynamically convex and global surface of section in $L(p,p-1)$ from the viewpoint of ECH
本文提出用嵌入接触同调（ECH）这一替代方法，在动力凸的$(L(p,p-1),ξ_{\mathrm{std}})$中构造圆盘型全局截面，并将其周期与第一个ECH谱建立联系。
We construct the Ruelle invariant of a volume preserving flow and a symplectic cocycle in any dimension and prove several properties. In the special case of the linearized Reeb flow on the boundary of a convex domain $X$ in $\mathbb{R}^{2n}$, we prove that the Ruelle invariant $\text{Ru}(X)$, the period of the systole $c(X)$ and the volume $\text{vol}{X}$ satisfy \[\text{Ru}(X) \cdot c(X) \le C(n) \cdot \text{vol}{X}\] Here $C(n) > 0$ is an explicit constant dependent on $n$. As an application, we construct dynamically convex contact forms on $S^{2n-1}$ that are not convex, disproving the equi…
Strong Arnold chord conjecture via normalized capacities
本文利用归一化容量证明，动态凸环面区域附近的所有星形区域边界上的任意 Legendrian 纽结均具有端点不同的 Reeb 弦，并给出高维推广。
Disk-like surfaces of section and symplectic capacities
本文证明R^4中动态凸区域的柱容量等于其边界Reeb流圆盘状全局截面的最小辛面积，并对C^3接近圆球的凸区域证明强Viterbo猜想，推广了已有收缩不等式。
Quantitative conditions for right-handedness of flows
本研究给出3-球面上动态凸Reeb流右手性的数值条件，找到显式常数δ_*<0.7225，证明2-球面上δ>δ_*的δ-pinched黎曼度量的测地流提升为右手流，其周期轨道绑定页为全局截面曲面的开书。
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。
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