arXiv subjects
Publications and source records attributed to Agustin Moreno.
This document collects contributions to the Open Problem List in Billiards and Quantitative Symplectic Geometry, compiled following discussions during the workshop ``Billiards and quantitative symplectic geometry'' that took place at the University of Heidelberg on July 14--18, 2025.
The goal of this note is to explore, from a geometric and probabilistic point of view, the dynamics of cone structures adapted to open book decompositions. This is inspired by the picture which arises in the study of the circular restricted three body problem (CR3BP). This yields geometric obstructions to reaching a point from another point in the phase space of the CR3BP.
We prove a higher-dimensional version of the well-known Poincar\'e--Birkhoff theorem, using Floer homology. We also prove a relative version for Lagrangian submanifolds. The motivation is finding periodic orbits and Hamiltonian chords in the circular, restricted three-body problem.
In this note, we show there exist infinitely many trajectories which are bi-normal (i.e. normal at initial and final times) to the xz-plane, in the Spatial Circular Restricted Three-Body Problem, for energies below or slightly above the first critical value and near the primaries, under the assumption of the twist condition as defined by Moreno-van-Koert in arXiv:2011.06562. This is an application of the relative Poincar\'e-Birkhoff theorem for Lagrangians in Liouville domains, as proven by the authors in arXiv:2408.06919.
In arXiv:2011.06562, the first author and Otto van Koert proved a generalized version of the classical Poincar\'e-Birkhoff theorem, for Liouville domains of any dimension. In this article, we prove a relative version for Lagrangians with Legendrian boundary. This gives interior chords of arbitrary large length, provided the twist condition introduced in arXiv:2011.06562 is satisfied. The motivation comes from finding spatial consecutive collision orbits of arbitrary large length in the spatial circular restricted three-body problem, which are relevant for gravitational assist in the context of orbital mechanics. This is an application of a local version of wrapped Floer homology, which we introduce as the open string analogue of local Floer homology for closed strings.
Using methods from symplectic geometry, the second and fifth authors have provided theoretical groundwork and tools aimed at analyzing periodic orbits, their stability and their bifurcations in families, for the purpose of space mission design. The Broucke stability diagram was refined, and the "Floer numerical invariants" where considered, as numbers which stay invariant before and after a bifurcation, and therefore serve as tests for the algorithms used. These tools were later employed for numerical studies. In this article, we will further illustrate these methods with numerical studies of families of orbits for the Jupiter-Europa and Saturn-Enceladus systems, with emphasis on planar-to-spatial bifurcations, from deformation of the families in Hill's lunar problem studied by the first author. We will also provide an algorithm for the numerical computation of Conley--Zehnder indices, which are instrumental in practice for determining which families of orbits connect to which. As an application, we use our tools to study a family of periodic orbits that approaches Enceladus at an altitude of 29km, and therefore may be used in future space missions to visit the water plumes.
We address the general problem of studying linear stability and bifurcations of periodic orbits for Hamiltonian systems of arbitrary degrees of freedom. We study the topology of the GIT sequence introduced by the first author and Urs frauenfelder, in arbitrary dimension. In particular, we note that the combinatorics encoding the linear stability of periodic orbits is governed by a quotient of the associahedron. Our approach gives a topological/combinatorial proof of the classical Krein--Moser theorem, and refines it for the case of symmetric orbits.
In this article, we extend the methods from arXiv:2011.06568, where the five dimensional analogue of the three dimensional finite energy foliations introduced by Hofer--Wysocki--Zehnder was identified, to the case where there the underlying (IP) contact $5$-fold admits a $6$-dimensional (IP) symplectic filling. We show that the filling induces a moduli space of pseudo-holomorphic curves which is itself a symplectic filling of the standard $3$-sphere, and hence symplectomorphic to the $4$-dimensional ball. We further show that whenever the contact form on the $5$-fold is strictly convex, then this moduli space is a strictly convex domain, so that the induced Reeb dynamics at the boundary is in particular dynamically convex. For the circular restricted three-body problem, this implies that whenever the spatial dynamics (near one of the heavy masses) is strictly convex, the holomorphic shadow of arXiv:2011.06568 is dynamically convex; this is shown to hold for near-integrable cases close to the Kepler problem, and with mass ratio either zero or sufficiently close to 1.
We extend the hierarchy functors of [33] to the case of strong symplectic cobordisms, via deformations with Maurer--Cartan elements. In particular, we prove that the concave boundary of a strong cobordism has finite algebraic planar torsion if the convex boundary does, which yields a functorial proof of finite algebraic planar torsion for contact manifolds admitting strong cobordisms to overtwisted contact manifolds. We also show the existence of contact $3$-folds without strong cobordisms to the standard contact $3$-sphere, that are not cofillable. We also include generalizations of the theory relating our notion of algebraic planar torsion to Latschev--Wendl's notion of algebraic torsion, discussing variations from counting holomorphic curves with general constraints and invariants extracted from higher genera holomorphic curves from an algebraic perspective.
The intention of this article is to illustrate the use of methods from symplectic geometry for practical purposes. Our intended audience is scientists interested in orbits of Hamiltonian systems (e.g. the three-body problem). The main directions pursued in this article are: (1) given two periodic orbits, decide when they can be connected by a regular family; (2) use numerical invariants from Floer theory which help predict the existence of orbits in the presence of a bifurcation; (3) attach a sign +/- to each elliptic or hyperbolic Floquet multiplier of a closed symmetric orbit, generalizing the classical Krein--Moser sign to also include the hyperbolic case; and (4) do all of the above in a visual, easily implementable and resource-efficient way. The mathematical framework is provided by the first and third authors, where the ``Broucke stability diagram'' was rediscovered, but further refined with the above signs, and algebraically reformulated in terms of GIT quotients of the symplectic group. The advantage of the above framework is that it applies to the study of closed orbits of an arbitrary Hamiltonian system. Moreover, in the case where the system admits symmetries in the form of ``reflections'', i.e. anti-symplectic involutions, which is the case for many systems of interest, the information provided for orbits which are symmetric is richer, and one may distinguish more symmetric orbits. This is the case for several well-known families in the space mission design industry, such as the Halo orbits, which are ubiquitous in real-life space missions. We will carry out numerical work based on the cell-mapping method, for the Jupiter-Europa and the Saturn-Enceladus systems. These are currently systems of interest, falling in the agenda of space agencies like NASA, as these icy moons are considered candidates for harbouring conditions suitable for extraterrestrial life.
In this article, for Hamiltonian systems with two degrees of freedom, we study doubly symmetric periodic orbits, i.e. those which are symmetric with respect to two (distinct) commuting antisymplectic involutions. These are ubiquitous in several problems of interest in mechanics. We show that, in dimension four, doubly symmetric periodic orbits cannot be negative hyperbolic. This has a number of consequences: (1) all covers of doubly symmetric orbits are good, in the sense of Symplectic Field Theory; (2) a non-degenerate doubly symmetric orbit is stable if and only if its CZ-index is odd; (3) a doubly symmetric orbit does not undergo period doubling bifurcation; and (4) there is always a stable orbit in any collection of doubly symmetric periodic orbits with negative SFT-Euler characteristic (as coined by the authors in arXiv 2206.00627). The above results follow from: (5) a symmetric orbit is negative hyperbolic if and only its two B-signs (introduced by the authors in arXiv 2109.09147) differ.
A smooth Anosov flow on a closed oriented three manifold $M$ gives rise to a Liouville structure on the four manifold $[-1,1]\times M$ which is not Weinstein, by a construction of Mitsumatsu and Hozoori. We call it the associated Anosov Liouville domain. It is well defined up to homotopy and only depends on the homotopy class of the original Anosov flow; its symplectic invariants are then invariants of the flow. We study the symplectic geometry of Anosov Liouville domains, via the wrapped Fukaya category, which we expect to be a powerful invariant of Anosov flows. The Lagrangian cylinders over the simple closed orbits span a natural $A_\infty$-subcategory, the orbit category of the flow. We show that it does not satisfy Abouzaid's generation criterion; it is moreover "very large", in the sense that is not split-generated by any strict sub-family. This is in contrast with the Weinstein case, where critical points of a Morse function play the role of the orbits. For the domain corresponding to the suspension of a linear Anosov diffeomorphism on the torus, we show that there are no closed exact Lagrangians which are either orientable, projective planes or Klein bottles. By contrast, in the case of the geodesic flow on a hyperbolic surface of genus $g \geq 2$ (corresponding to the McDuff example), we construct an exact Lagrangian torus for each embedded closed geodesic, thus obtaining at least $3g-3$ tori which are not Hamiltonian isotopic to each other. For these two prototypical cases of Anosov flows, we explicitly compute the symplectic cohomology of the associated domains, as well as the wrapped Floer cohomology of the Lagrangian cylinders, and several pair-of-pants products.
We show that for all $n \ge 3$, any $(2n+1)$-dimensional manifold that admits a tight contact structure, also admits a tight but non-fillable contact structure, in the same almost contact class. For $n=2$, we obtain the same result, provided that the first Chern class vanishes. We further construct Liouville but not Weinstein fillable contact structures on any Weinstein fillable contact manifold of dimension at least $7$ with torsion first Chern class.
In this article, we identify the 5-dimensional analogue of the finite energy foliations introduced by Hofer--Wysocki--Zehnder for the study of 3-dimensional Reeb flows, and show that these exist for the spatial circular restricted three-body problem (SCR3BP) whenever the planar dynamics is convex. We introduce the notion of a fiberwise-recurrent point, which may be thought of as a symplectic version of the leafwise intersections introduced by Moser, and show that they exist in abundance for a perturbative regime in the SCR3BP. We then use this foliation to induce a Reeb flow on the standard 3-sphere, via the use of pseudo-holomorphic curves, to be understood as the best approximation of the given dynamics that preserves the foliation. We discuss examples, further geometric structures, and speculate on possible applications.
Given a contact structure on a manifold $V$ together with a supporting open book decomposition, Bourgeois gave an explicit construction of a contact structure on $V \times \mathbb{T}^2$. We prove that all such structures are universally tight in dimension $5$, independent on whether the original contact manifold is itself tight or overtwisted. In arbitrary dimensions, we provide obstructions to the existence of strong symplectic fillings of Bourgeois manifolds. This gives a broad class of new examples of weakly but not strongly fillable contact $5$-manifolds, as well as the first examples of weakly but not strongly fillable contact structures in all odd dimensions. These obstructions are particular instances of more general obstructions for $\mathbb S^1$-invariant contact manifolds. We also obtain a classification result in arbitrary dimensions, namely that the unit cotangent bundle of the $n$-torus has a unique symplectically aspherical strong filling up to diffeomorphism.
We propose a contact-topological approach to the spatial circular restricted three-body problem, for energies below and slightly above the first critical energy value. We prove the existence of a circle family of global hypersurfaces of section for the regularized dynamics. Below the first critical value, these hypersurfaces are diffeomorphic to the unit disk cotangent bundle of the $2$-sphere, and they carry symplectic forms on their interior, which are each deformation equivalent to the standard symplectic form. The boundary of the global hypersurface of section is an invariant set for the regularized dynamics that is equal to a level set of the Hamiltonian describing the regularized planar problem. The first return map is Hamiltonian, and restricts to the boundary as the time-$1$ map of a positive reparametrization of the Reeb flow in the planar problem. This construction holds for any choice of mass ratio, and is therefore non-perturbative. We illustrate the technique in the completely integrable case of the rotating Kepler problem, where the return map can be studied explicitly.
We provide topological obstructions to the existence of orbit cylinders of symmetric orbits, for mechanical systems preserved by antisymplectic involutions (e.g. the restricted three-body problem). Such cylinders induce continuous paths which do not cross the bifurcation locus of suitable GIT quotients of the symplectic group, which are branched manifolds whose topology provide the desired obstructions. Namely, the complement of the corresponding loci consist of several connected components which we enumerate and explicitly describe; by construction these cannot be joined by a path induced by an orbit cylinder. Our construction extends the notions from Krein theory (which only applies for elliptic orbits), to allow also for the case of symmetric orbits which are hyperbolic. This gives a general theoretical framework for the study of stability and bifurcations of symmetric orbits, with a view towards practical and numerical implementations within the context of space mission design. We shall explore this in upcoming work.
We prove a generalization of the classical Poincaré--Birkhoff theorem for Liouville domains, in arbitrary even dimensions. This is inspired by the existence of global hypersurfaces of section for the spatial case of the restricted three-body problem (as proved by the authors in arXiv:2011.10386).