Global Hamiltonian dynamics on singular symplectic manifolds

In this thesis, we study the Reeb and Hamiltonian dynamics on singular symplectic and contact manifolds. Those structures are motivated by singularities coming from classical mechanics and fluid dynamics. We start by studying generalized contact structures where the non-integrability condition fails on a hypersurface, the critical hypersurface. Those structures, called $b$-contact structures, arise from hypersurfaces in $b$-symplectic manifolds that have been previously studied extensively in the past. Formerly, this odd-dimensional counterpart to $b$-symplectic geometry has been neglected in the existing vast literature. Examples are given and local normal forms are proved. The local geometry of those manifolds is examined using the language of Jacobi manifolds, which provides an adequate set-up and leads to understanding the geometric structure on the critical hypersurface. We further consider other types of singularities in contact geometry, as for instance higher order singularities, called $b^m$-contact forms, or singularities of folded type. Obstructions to the existence of those structures are studied and the topology of $b^m$-contact manifolds is related to the existence of convex contact hypersurfaces and further relations to smooth contact structures are described using the desingularization technique. We continue examining the dynamical properties of the Reeb vector field associated to a given $b^m$-contact form. The relation of those structures to celestial mechanics underlines the relevance for existence results of periodic orbits of the Hamiltonian vector field in the $b^m$-symplectic setting and Reeb vector fields for $b^m$-contact manifolds. In this light, we prove that in dimension $3$, there are always infinitely many periodic Reeb orbits on the critical surface, but describe examples without periodic orbits away from it in any dimension. We prove that there are traps for this vector field and discuss possible extensions to prove the existence of plugs. We will see that in the case of overtwisted disks away from the critical hypersurface and some additional conditions, Weinstein conjecture holds: more precisely there exists either a periodic Reeb orbit away from the critical hypersurface or a $1$-parametric family in the neighbourhood of it. The mentioned results shed new light towards a singular version for this conjecture. The obtained results are applied to the particular case of the restricted planar circular three body problem, where we prove that after the McGehee change, there are infinitely many non-trivial periodic orbits at the manifold at infinity for positive energy values.En esta tesis, estudiamos la dinámica de Reeb y Hamiltoniana en variedades simplécticas y de contacto con singularidades. El estudio de estas variedades está motivado por singularidades que tienen su origen en la mecánica clásica y la dinámica de fluidos. Empezamos estudiando una generalización de las estructuras de contacto, en la cual la condición de no integrabilidad falla en una hipersuperficie, llamada la hipersuperficie crítica. Estas estructuras geométricas, llamadas estructuras de $b$-contacto, surgen de hipersuperficies en variedades $b$-simplécticas, estudiadas en el pasado. Hasta el momento, este equivalente de dimensión impar de la geometría $b$-simpléctica ha sido desatendido en la literatura existente. Después de los primeros ejemplos, probamos la existencia de formas locales. Estudiamos la geometría local de estas variedades usando el lenguaje de variedades de Jacobi, que resultan ser técnicas adecuadas para entender la estructura geométrica en la hipersuperficie crítica. Consideramos también singularidades de orden superior, formas de $b^m$-contacto, y singularidades de tipo folded. Continuamos con el estudio de las obstrucciones a la existencia de estas estructuras y relacionamos la topología de variedades de $b^m$-contacto con la existencia de hipersuperficies convexas. Describimos relaciones entre formas de $b^m$-contacto y formas de contacto diferenciables usando técnicas de desingularización. Examinamos las propiedades del campo de Reeb asociado a una forma de $b^m$-contacto dada. La relación de estas estructuras con la mecánica celeste pone en relieve la importancia del estudio de órbitas periódicas de este campo vectorial. Comprobamos que, en dimensión $3$, el campo de Reeb en la hipersuperficie crítica admite infinitas órbitas periódicas. Sin embargo, describimos ejemplos sin órbitas periódicas fuera de la hipersuperficie crítica en cualquier dimensión. Comprobamos la existencia de traps y discutimos la posible existencia de plugs. En el caso de un disco \emph{overtwisted} fuera de la hipersuperficie se satisface la conjetura de Weinstein: en concreto, o bien existe una órbita periódica de Reeb fuera de la hipersuperficie de contacto o bien existe una familia de órbitas periódicas en un entorno de la hipersuperficie. Estos resultados sugieren una versión singular de dicha conjetura. Aplicamos los resultados obtenidos al caso del problema de los tres cuerpos restringido circular: comprobamos que después del cambio de coordenadas de McGehee, existen infinitas órbitas periódicas en la variedad en el infinito para valores positivos de la energía.Postprint (published version

Existence and classification of bb-contact structures

A $b$-contact structure on a $b$-manifold $(M,Z)$ is a singular Jacobi structure on $M$ satisfying a transversality condition along the hypersurface $Z$. We show that, in three dimensions, $b$-contact structures with overtwisted three-dimensional leaves satisfy an existence $h$-principle that allows prescribing the induced singular foliation. We give a method to classify $b$-contact structures on a given $b$-manifold and use it to give a classification on $S^3$ with either a two-sphere or an unknotted torus as the critical surface. We also discuss generalizations to higher dimensions.Comment: 27 pages. Minor correction

Contact structures with singularities

We study singular contact structures, which are tangent to a given smooth hypersurface Z and satisfy certain transversality conditions. These singular contact structures are determined by the kernel of non-smooth differential forms, called bm-contact forms having an associated critical hypersurface Z. We provide several constructions, prove local normal forms and study the induced structure on the critical hypersurface. In the last section of this paper we tackle the problem of existence of bm-contact structures on a given manifold. We prove that convex hypersurfaces can be realized as critical set of b2k-contact structures. In particular, in the 3-dimensional case, this construction yields the existence of a generic set of surfaces Z such that the pair (M;Z) is a b2k-contact manifold and Z is its critical hypersurfacePreprin

An Invitation to Singular Symplectic Geometry

In this paper we analyze in detail a collection of motivating examples to consider $b^m$-symplectic forms and folded-type symplectic structures. In particular, we provide models in Celestial Mechanics for every $b^m$-symplectic structure. At the end of the paper, we introduce the odd-dimensional analogue to $b$-symplectic manifolds: $b$-contact manifolds.Comment: 14 pages, 1 figur

Contact structures with singularities: from local to global

In this article we introduce and analyze in detail singular contact structures, with an emphasis on bm -contact structures, which are tangent to a given smooth hypersurface Z and satisfy certain transversality conditions. These singular contact structures are determined by the kernel of non-smooth differential forms, called bm -contact forms, having an associated critical hypersurface Z. We provide several constructions, prove local normal forms, and study the induced structure on the critical hypersurface. The topology of manifolds endowed with such singular contact forms are related to smooth contact structures via desingularization. The problem of existence of bm -contact structures on a given manifold is also tackled in this paper. We prove that a connected component of a convex hypersurface of a contact manifold can be realized as a connected component of the critical set of a bm -contact structure. In particular, given an almost contact manifold M with a hypersurface Z, this yields the existence of a b2k -contact structure on M realizing Z as a critical set. As a consequence of the desingularization techniques in [21], we prove the existence of folded contact forms on any almost contact manifold.Eva Miranda and Cédric Oms are partially supported by the AEI grant PID2019-103849GB-I00 of MCIN/AEI/10.13039/501100011033Peer ReviewedObjectius de Desenvolupament Sostenible::14 - Vida SubmarinaObjectius de Desenvolupament Sostenible::13 - Acció per al ClimaPostprint (published version

On the singular Weinstein conjecture and the existence of escape orbits for bb-Beltrami fields

Motivated by Poincar\'e's orbits going to infinity in the (restricted) three-body (see [26] and [6]), we investigate the generic existence of heteroclinic-like orbits in a neighbourhood of the critical set of a $b$-contact form. This is done by using the singular counterpart [3] of Etnyre--Ghrist's contact/Beltrami correspondence [9], and genericity results concerning eigenfunctions of the Laplacian established by Uhlenbeck [29]. Specifically, we analyze the $b$-Beltrami vector fields on $b$-manifolds of dimension $3$ and prove that for a generic asymptotically exact $b$-metric they exhibit escape orbits. We also show that a generic asymptotically symmetric $b$-Beltrami vector field on an asymptotically flat $b$-manifold has a generalized singular periodic orbit and at least $4$ escape orbits. Generalized singular periodic orbits are trajectories of the vector field whose $\alpha$- and $\omega$-limit sets intersect the critical surface. These results are a first step towards proving the singular Weinstein conjecture.Comment: 18 pages, 2 figures, minor change

On the singular Weinstein conjecture and the existence of escape orbits for b-Beltrami fields

Motivated by Poincare’s orbits going to infinity in the (restricted) three-body problem ´ (see [29] and [7]), we investigate the generic existence of heteroclinic-like orbits in a neighbourhood of the critical set of a b-contact form. This is done by using a singular counterpart [4] of Etnyre– Ghrist’s contact/Beltrami correspondence [11], and genericity results concerning eigenfunctions of the Laplacian established by Uhlenbeck [33]. Specifically, we analyze the b-Beltrami vector fields on b-manifolds of dimension 3 and prove that for a generic asymptotically exact b-metric they exhibit escape orbits. We also show that a generic asymptotically symmetric b-Beltrami vector field on an asymptotically flat b-manifold has a generalized singular periodic orbit and at least 4 escape orbits. Generalized singular periodic orbits are trajectories of the vector field whose a- and ¿-limit sets intersect the critical surface. These results are a first step towards proving the singular Weinstein conjecture.E. M. is supported by the Catalan Institution for Research and Advanced Studies via an ICREA Academia Prize 2016. Eva Miranda and Cedric Oms are supported by the grants reference number MTM2015-69135-P (MINECO/FEDER) ánd reference number 2017SGR932 (AGAUR) and the project PID2019-103849GB-I00 / AEI / 10.13039/501100011033. C. O. has been supported by an FNR-AFR PhD predoctoral grant (project GLADYSS) until October 2nd, 2020 and by a SECTI-Postdoctoral grant financed by Eva Miranda’s ICREA Academia immediately after. D. P.-S. is supported by the grants MTM PID2019-106715GB-C21 (MICINN) and Europa Excelencia EUR2019- 103821 (MCIU). This work is supported in part by the ICMAT–Severo Ochoa grant SEV-2015-0554 and the CSIC grant 20205CEX001.Preprin

2N or infinitely many escape orbits

In this short note, we prove that singular Reeb vector fields associated with generic Melrose-type $b$-contact forms have either (at least) $2N$ or an infinite number of escape orbits, where $N$ denotes the number of connected components of the critical set. We obtain this result as a corollary of the same statement for the number of escape orbits of singular Beltrami fields using the singular version of Etnyre-Ghrist's contact/Beltrami correspondence.Comment: 12 pages, for Alain Chenciner on his 2N-birthda

Global Hamiltonian dynamics on singular symplectic manifolds

In this thesis, we study the Reeb and Hamiltonian dynamics on singular symplectic and contact manifolds. Those structures are motivated by singularities coming from classical mechanics and fluid dynamics. We start by studying generalized contact structures where the non-integrability condition fails on a hypersurface, the critical hypersurface. Those structures, called $b$-contact structures, arise from hypersurfaces in $b$-symplectic manifolds that have been previously studied extensively in the past. Formerly, this odd-dimensional counterpart to $b$-symplectic geometry has been neglected in the existing vast literature. Examples are given and local normal forms are proved. The local geometry of those manifolds is examined using the language of Jacobi manifolds, which provides an adequate set-up and leads to understanding the geometric structure on the critical hypersurface. We further consider other types of singularities in contact geometry, as for instance higher order singularities, called $b^m$-contact forms, or singularities of folded type. Obstructions to the existence of those structures are studied and the topology of $b^m$-contact manifolds is related to the existence of convex contact hypersurfaces and further relations to smooth contact structures are described using the desingularization technique. We continue examining the dynamical properties of the Reeb vector field associated to a given $b^m$-contact form. The relation of those structures to celestial mechanics underlines the relevance for existence results of periodic orbits of the Hamiltonian vector field in the $b^m$-symplectic setting and Reeb vector fields for $b^m$-contact manifolds. In this light, we prove that in dimension $3$, there are always infinitely many periodic Reeb orbits on the critical surface, but describe examples without periodic orbits away from it in any dimension. We prove that there are traps for this vector field and discuss possible extensions to prove the existence of plugs. We will see that in the case of overtwisted disks away from the critical hypersurface and some additional conditions, Weinstein conjecture holds: more precisely there exists either a periodic Reeb orbit away from the critical hypersurface or a $1$-parametric family in the neighbourhood of it. The mentioned results shed new light towards a singular version for this conjecture. The obtained results are applied to the particular case of the restricted planar circular three body problem, where we prove that after the McGehee change, there are infinitely many non-trivial periodic orbits at the manifold at infinity for positive energy values.En esta tesis, estudiamos la dinámica de Reeb y Hamiltoniana en variedades simplécticas y de contacto con singularidades. El estudio de estas variedades está motivado por singularidades que tienen su origen en la mecánica clásica y la dinámica de fluidos. Empezamos estudiando una generalización de las estructuras de contacto, en la cual la condición de no integrabilidad falla en una hipersuperficie, llamada la hipersuperficie crítica. Estas estructuras geométricas, llamadas estructuras de $b$-contacto, surgen de hipersuperficies en variedades $b$-simplécticas, estudiadas en el pasado. Hasta el momento, este equivalente de dimensión impar de la geometría $b$-simpléctica ha sido desatendido en la literatura existente. Después de los primeros ejemplos, probamos la existencia de formas locales. Estudiamos la geometría local de estas variedades usando el lenguaje de variedades de Jacobi, que resultan ser técnicas adecuadas para entender la estructura geométrica en la hipersuperficie crítica. Consideramos también singularidades de orden superior, formas de $b^m$-contacto, y singularidades de tipo folded. Continuamos con el estudio de las obstrucciones a la existencia de estas estructuras y relacionamos la topología de variedades de $b^m$-contacto con la existencia de hipersuperficies convexas. Describimos relaciones entre formas de $b^m$-contacto y formas de contacto diferenciables usando técnicas de desingularización. Examinamos las propiedades del campo de Reeb asociado a una forma de $b^m$-contacto dada. La relación de estas estructuras con la mecánica celeste pone en relieve la importancia del estudio de órbitas periódicas de este campo vectorial. Comprobamos que, en dimensión $3$, el campo de Reeb en la hipersuperficie crítica admite infinitas órbitas periódicas. Sin embargo, describimos ejemplos sin órbitas periódicas fuera de la hipersuperficie crítica en cualquier dimensión. Comprobamos la existencia de traps y discutimos la posible existencia de plugs. En el caso de un disco \emph{overtwisted} fuera de la hipersuperficie se satisface la conjetura de Weinstein: en concreto, o bien existe una órbita periódica de Reeb fuera de la hipersuperficie de contacto o bien existe una familia de órbitas periódicas en un entorno de la hipersuperficie. Estos resultados sugieren una versión singular de dicha conjetura. Aplicamos los resultados obtenidos al caso del problema de los tres cuerpos restringido circular: comprobamos que después del cambio de coordenadas de McGehee, existen infinitas órbitas periódicas en la variedad en el infinito para valores positivos de la energía

Global Hamiltonian dynamics on singular symplectic manifolds

In this thesis, we study the Reeb and Hamiltonian dynamics on singular symplectic and contact manifolds. Those structures are motivated by singularities coming from classical mechanics and fluid dynamics. We start by studying generalized contact structures where the non-integrability condition fails on a hypersurface, the critical hypersurface. Those structures, called $b$-contact structures, arise from hypersurfaces in $b$-symplectic manifolds that have been previously studied extensively in the past. Formerly, this odd-dimensional counterpart to $b$-symplectic geometry has been neglected in the existing vast literature. Examples are given and local normal forms are proved. The local geometry of those manifolds is examined using the language of Jacobi manifolds, which provides an adequate set-up and leads to understanding the geometric structure on the critical hypersurface. We further consider other types of singularities in contact geometry, as for instance higher order singularities, called $b^m$-contact forms, or singularities of folded type. Obstructions to the existence of those structures are studied and the topology of $b^m$-contact manifolds is related to the existence of convex contact hypersurfaces and further relations to smooth contact structures are described using the desingularization technique. We continue examining the dynamical properties of the Reeb vector field associated to a given $b^m$-contact form. The relation of those structures to celestial mechanics underlines the relevance for existence results of periodic orbits of the Hamiltonian vector field in the $b^m$-symplectic setting and Reeb vector fields for $b^m$-contact manifolds. In this light, we prove that in dimension $3$, there are always infinitely many periodic Reeb orbits on the critical surface, but describe examples without periodic orbits away from it in any dimension. We prove that there are traps for this vector field and discuss possible extensions to prove the existence of plugs. We will see that in the case of overtwisted disks away from the critical hypersurface and some additional conditions, Weinstein conjecture holds: more precisely there exists either a periodic Reeb orbit away from the critical hypersurface or a $1$-parametric family in the neighbourhood of it. The mentioned results shed new light towards a singular version for this conjecture. The obtained results are applied to the particular case of the restricted planar circular three body problem, where we prove that after the McGehee change, there are infinitely many non-trivial periodic orbits at the manifold at infinity for positive energy values.En esta tesis, estudiamos la dinámica de Reeb y Hamiltoniana en variedades simplécticas y de contacto con singularidades. El estudio de estas variedades está motivado por singularidades que tienen su origen en la mecánica clásica y la dinámica de fluidos. Empezamos estudiando una generalización de las estructuras de contacto, en la cual la condición de no integrabilidad falla en una hipersuperficie, llamada la hipersuperficie crítica. Estas estructuras geométricas, llamadas estructuras de $b$-contacto, surgen de hipersuperficies en variedades $b$-simplécticas, estudiadas en el pasado. Hasta el momento, este equivalente de dimensión impar de la geometría $b$-simpléctica ha sido desatendido en la literatura existente. Después de los primeros ejemplos, probamos la existencia de formas locales. Estudiamos la geometría local de estas variedades usando el lenguaje de variedades de Jacobi, que resultan ser técnicas adecuadas para entender la estructura geométrica en la hipersuperficie crítica. Consideramos también singularidades de orden superior, formas de $b^m$-contacto, y singularidades de tipo folded. Continuamos con el estudio de las obstrucciones a la existencia de estas estructuras y relacionamos la topología de variedades de $b^m$-contacto con la existencia de hipersuperficies convexas. Describimos relaciones entre formas de $b^m$-contacto y formas de contacto diferenciables usando técnicas de desingularización. Examinamos las propiedades del campo de Reeb asociado a una forma de $b^m$-contacto dada. La relación de estas estructuras con la mecánica celeste pone en relieve la importancia del estudio de órbitas periódicas de este campo vectorial. Comprobamos que, en dimensión $3$, el campo de Reeb en la hipersuperficie crítica admite infinitas órbitas periódicas. Sin embargo, describimos ejemplos sin órbitas periódicas fuera de la hipersuperficie crítica en cualquier dimensión. Comprobamos la existencia de traps y discutimos la posible existencia de plugs. En el caso de un disco \emph{overtwisted} fuera de la hipersuperficie se satisface la conjetura de Weinstein: en concreto, o bien existe una órbita periódica de Reeb fuera de la hipersuperficie de contacto o bien existe una familia de órbitas periódicas en un entorno de la hipersuperficie. Estos resultados sugieren una versión singular de dicha conjetura. Aplicamos los resultados obtenidos al caso del problema de los tres cuerpos restringido circular: comprobamos que después del cambio de coordenadas de McGehee, existen infinitas órbitas periódicas en la variedad en el infinito para valores positivos de la energía