arXiv subjects
Publications and source records attributed to Pedro A. S. Salomão.
We exhibit transverse knot types on the standard contact $3$-sphere that cannot be realized as periodic Reeb orbits of a dynamically convex contact form. In particular, such transverse knot types do not arise as closed characteristics of strictly convex energy levels on a four dimensional symplectic vector space.
We study symplectic embeddings between four-dimensional toric domains using embedded contact homology. Extending Hutchings' criterion for embeddings between convex toric domains, we obtain obstructions for embeddings from convex toric domains into concave toric domains, from semi-weakly convex toric domains into convex toric domains, and from concave toric domains into concave toric domains. The obstructions are formulated in terms of factorizations of convex, concave, and semi-weakly convex generators together with combinatorial constraints relating their ECH indices and actions. As applications, we recover the sharp obstruction for symplectic embeddings of a polydisk into a union of cylinders, and obtain sharp results for embeddings of certain quadrilateral toric domains into balls and ellipsoids. We also show that these obstructions are in some cases strictly stronger than those coming only from ECH capacities.
We prove Birkhoff's retrograde orbit conjecture in Hill's lunar problem by showing that the retrograde orbit bounds a disk-like global surface of section for every energy below the critical value. We also obtain a global description of the dynamics through the critical energy level by constructing finite energy foliations for energies slightly above it. The binding of these foliations consists of the retrograde orbit together with the Lyapunov orbits near the critical points. As a consequence, there exist infinitely many periodic orbits and infinitely many trajectories asymptotic to the Lyapunov orbits. The proof combines pseudo-holomorphic curve techniques with a new convexity theorem for Hill's lunar problem. More precisely, we construct an explicit global symplectic change of coordinates under which the bounded regularized component becomes strictly convex up to the critical energy level. This convexity implies strong dynamical consequences, including lower bounds for the Conley-Zehnder indices of periodic orbits, and allows the application of the Hofer-Wysocki-Zehnder theory of finite energy foliations. As a result, we obtain disk-like global surfaces of section below the critical level and $2-3-2$ foliations for energies slightly above it.
We establish a general criterion for the existence of finite energy foliations on contact three-manifolds with boundary, imposing strong global constraints on the associated Reeb flows. Our main abstract result shows that a configuration of periodic orbits, consisting of hyperbolic boundary orbits of Conley-Zehnder index $2$ and an interior orbit of index at least $3$, gives rise to a finite energy foliation, provided that no additional contractible periodic orbit satisfies a specific rotation, linking, and action condition. This identifies the precise dynamical obstruction to the existence of such foliations. These foliations organize the flow in a regime where the dynamics is typically chaotic, and imply the existence of infinitely many periodic orbits and infinitely many homoclinic/heteroclinic orbits to the Lyapunov orbits. We apply this result to the circular planar restricted three-body problem. For mass ratios sufficiently close to $\frac{1}{2}$ and energies slightly above the first Lagrange value, the regularized energy surface $\mathbb{R}P^3 \# \mathbb{R}P^3$ admits a finite energy foliation whose binding consists of the retrograde orbits around the primaries together with the Lyapunov orbit near the first Lagrange point. Moreover, the convexity of the critical energy surface provides a proof of Birkhoff's retrograde orbit conjecture for mass ratios sufficiently close to $\frac{1}{2}$ and all energies below the first Lagrange value.
For each integer $m \geq 1$, we construct a finite-dimensional family of rational maps, given by Blaschke-type products, whose restriction to the unit circle consists of $2m$-multimodal maps. We show that every post-critically finite $2m$-multimodal circle map satisfying natural dynamical conditions is topologically conjugate to a map in this family. Moreover, we prove that this realization is unique up to rotation: two maps in the family that are topologically conjugate on the circle differ by a rigid rotation. In particular, the family provides a canonical model realizing all post-critically finite combinatorics in this class. The proofs combine a detailed description of the critical geometry of these Blaschke-type maps with a Thurston-type fixed point argument for a pull-back operator on the parameter space.
A contact form $λ$ on a closed contact three-manifold $(M,ξ)$ is called weakly convex if either it has no contractible Reeb orbit, or the first Chern class of $ξ$ vanishes on $π_2(M)$, and the index of every contractible Reeb orbit is at least $2$. We present conditions for a weakly convex contact form to admit a well-defined cylindrical contact homology. The key point is a cancellation mechanism for boundary degenerations involving index-2 Reeb orbits, based on a parity property of holomorphic planes.
We investigate the dynamics of a two-degree-of-freedom mechanical system for energies slightly above a critical value. The critical set of the potential function is assumed to contain a finite number of saddle points. As the energy increases across the critical value, a disk-like component of the Hill region gets connected to other components precisely at the saddles. Under certain convexity assumptions on the critical set, we show the existence of a weakly convex foliation in the region of the energy surface where the interesting dynamics takes place. The binding of the foliation is formed by the index-$2$ Lyapunov orbits in the neck region about the rest points and a particular index-$3$ orbit. Among other dynamical implications, the transverse foliation forces the existence of periodic orbits, homoclinics, and heteroclinics to the Lyapunov orbits. We apply the results to the Hénon-Heiles potential for energies slightly above $1/6$. We also discuss the existence of transverse foliations for decoupled mechanical systems, including the frozen Hill's lunar problem with centrifugal force, the Stark problem, the Euler problem of two centers, and the potential of a chemical reaction.
We study dynamical constraints arising from Embedded Contact Homology (ECH) in the spatial isosceles three-body problem. For energies below the critical level, the dynamics on the energy surface is identified with a Reeb flow on the tight three-sphere. We obtain quantitative estimates for the Euler orbit, including monotonicity of its transverse rotation number and a strict inequality comparing its action with the contact volume. Combined with the ECH classification of Reeb flows on the tight three-sphere with two simple periodic orbits, these estimates rule out the two-orbit scenario, thus forcing every compact energy surface below the critical level to have infinitely many periodic orbits. The result admits a dynamical interpretation via disk-like global surfaces of section bounded by the Euler orbit. In this setting, the rotation number and the contact volume define a non-trivial twist interval which encodes the relative winding of periodic orbits. For energies above the critical level, where the energy surface is non-compact, we prove the existence of infinitely many periodic orbits and infinitely many parabolic trajectories via twist estimates near infinity.
This paper is about the existence of periodic orbits near an equilibrium point of a two-degree-of-freedom Hamiltonian system. The equilibrium is supposed to be a nondegenerate minimum of the Hamiltonian. Every sphere-like component of the energy surface sufficiently close to the equilibrium contains at least two periodic orbits forming a Hopf link (A. Weinstein [19]). A theorem by Hofer, Wysocki, and Zehnder [9] implies that there are either precisely two or infinitely many periodic orbits on such a component of the energy surface. This multiplicity result follows from the existence of a disk-like global surface of section. If a certain non-resonance condition on the rotation numbers of the orbits of the Hopf link is satisfied [8], then infinitely many periodic orbits follow. This paper aims to present explicit conditions on the Birkhoff-Gustavson normal forms of the Hamiltonian function at the equilibrium point that ensure the existence of infinitely many periodic orbits on the energy surface by checking the non-resonance condition as in [8] and not making use of any global surface of section. The main results focus on strongly resonant equilibrium points and apply to the Spatial Isosceles Three-Body Problem, Hill's Lunar Problem, and the Hénon-Heiles System.
A contact form on the tight $3$-sphere $(S^3,ξ_0)$ is called weakly convex if the Conley-Zehnder index of every Reeb orbit is at least $2$. In this article, we study Reeb flows of weakly convex contact forms on $(S^3,ξ_0)$ admitting a prescribed finite set of index-$2$ Reeb orbits, which are all hyperbolic and mutually unlinked. We present conditions so that these index-$2$ orbits are binding orbits of a genus zero transverse foliation whose additional binding orbits have index $3$. In addition, we show in the real-analytic case that the topological entropy of the Reeb flow is positive if the branches of the stable/unstable manifolds of the index-$2$ orbits are mutually non-coincident.
Pseudo-holomorphic curves in symplectizations, as introduced by Hofer in 1993, and then developed by Hofer, Wysocki, and Zehnder, have brought new insights to Hamiltonian dynamics, providing new approaches to some classical questions in Celestial Mechanics. This short survey presents some recent developments in Reeb dynamics based on the theory of pseudo-holomorphic curves in symplectizations, focusing on transverse foliations near critical energy surfaces.
We study the spatial isosceles three-body problem from the perspective of Symplectic Dynamics. For certain choices of mass ratio, angular momentum, and energy, the dynamics on the energy surface is equivalent to a Reeb flow on the tight three-sphere. We find a Hopf link formed by the Euler orbit and a symmetric brake orbit, which spans an open book decomposition whose pages are annulus-like global surfaces of section. In the case of large mass ratios, the Hopf link is non-resonant, forcing the existence of infinitely many periodic orbits. The rotation number of the Euler orbit plays a fundamental role in the existence of periodic orbits and their symmetries. We explore such symmetries in the Hill region and show that the Euler orbit is negative hyperbolic for an open set of parameters while it can never be positive hyperbolic. Finally, we address convexity and determine for each parameter whether the energy surface is strictly convex, convex, or non-convex. Dynamical consequences of this fact are then discussed.
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.
We introduce numerical invariants of contact forms in dimension three and use asymptotic cycles to estimate them. As a consequence, we prove a version for Anosov Reeb flows of results due to Hutchings and Weiler on mean actions of periodic points. The main tool is the Action-Linking Lemma, expressing the contact area of a surface bounded by periodic orbits as the Liouville average of the asymptotic intersection number of most trajectories with the surface.
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.
We survey some recent developments in the quest for global surfaces of section for Reeb flows in dimension three using methods from Symplectic Topology. We focus on applications to geometry, including existence of closed geodesics and sharp systolic inequalities. Applications to topology and celestial mechanics are also presented.
We prove that the systolic ratio of a sphere of revolution $S$ does not exceed $π$ and equals $π$ if and only if $S$ is Zoll. More generally, we consider the rotationally symmetric Finsler metrics on a sphere of revolution which are defined by shifting the tangent unit circles by a Killing vector field. We prove that in this class of metrics the systolic ratio does not exceed $π$ and equals $π$ if and only if the metric is Riemannian and Zoll.
We study two-degree-of-freedom Hamiltonian systems. Let us assume that the zero energy level of a real-analytic Hamiltonian function $H:\mathbb{R}^4 \to \mathbb{R}$ contains a saddle-center equilibrium point lying in a strictly convex sphere-like singular subset $S_0\subset H^{-1}(0)$. From previous work [de Paulo-Salomão, Memoirs of the AMS] we know that for any small energy $E>0$, the energy level $H^{-1}(E)$ contains a closed $3$-ball $S_E$ in a neighborhood of $S_0$ admitting a singular foliation called $2-3$ foliation. One of the binding orbits of this singular foliation is the Lyapunoff orbit $P_{2,E}$ contained in the center manifold of the saddle-center. The other binding orbit lies in the interior of $S_E$ and spans a one parameter family of disks transverse to the Hamiltonian vector field. In this article we show that the $2-3$ foliation forces the existence of infinitely many periodic orbits and infinitely many homoclinics to $P_{2,E}$ in $S_E$. Moreover, if the branches of the stable and unstable manifolds of $P_{2,E}$ inside $S_E$ do not coincide then the Hamiltonian flow on $S_E$ has positive topological entropy. We also present applications of these results to some classical Hamiltonian systems.