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The version of record of this article, first published in Journal of fixed point theory and its applications, is available online at Publisher’s website: http://doi.org/10.1007/s11784-024-01105-y, In this article, we study the Hamiltonian dynamics on singular symplectic manifolds and prove the Arnold conjecture for a large class of bm-symplectic manifolds. Novel techniques are introduced to associate smooth symplectic forms to the original singular symplectic structure, under some mild conditions. These techniques yield the validity of the Arnold conjecture for singular symplectic manifolds across multiple scenarios. More precisely, we prove a lower bound on the number of 1-periodic Hamiltonian orbits for b2m-symplectic manifolds depending only on the topology of the manifold. Moreover, for bm-symplectic surfaces, we improve the lower bound depending on the topology of the pair (M, Z). We then venture into the study of Floer homology to this singular realm and we conclude with a list of open questions., Funding OpenAccess funding provided thanks to the CRUE-CSIC agreement with Springer Nature. All the authors are partially supported by the Spanish State Research Agency grant PID2019-103849GB-I00 of MICIU/AEI/10.13039/50110 00110331934 and by the AGAUR project 2021 SGR 00603. J. Brugu´ es is supported by the FWO-FNRS Excellence of Science project G0H4518N “Symplectic techniques in differential geometry” with UA Antigoon Project-ID 36584.E. Miranda is supported by the Catalan Institution for Research and Advanced Studies via an ICREA Academia Prize 2021 and by the Alexander Von Humboldt foundation via a Friedrich Wilhelm Bessel Research Award. E. Miranda is also supported by the Spanish State Research Agency, through the 16 Page 52 of 60 J. Brugués et al. Severo Ochoa and Mar´ ıa de Maeztu Program for Centers and Units of Excellence in R&D (Project CEX2020-001084-M). C. Oms acknowledges financial support from the Juan de la Cierva postdoctoral grant FJC2021-046811-I / MICIU/AEI /10.13039/501100011033 and by the European Union NextGenerationEU/PRTR., Peer Reviewed, Postprint (published version)

In this article, we study the Hamiltonian dynamics on singular symplectic manifolds and prove the Arnold conjecture for a large class of bm-symplectic manifolds. More precisely, we prove a lower bound on the number 1-periodic Hamiltonian orbits for b2m-symplectic manifolds depending only on the topology of the manifold. Moreover, for bm-symplectic surfaces, we improve the lower bound depending on the topology of the pair (M, Z). We then venture into the study of Floer homology to this singular realm and we conclude with a list of open questions., Preprint

Postprint (author's final draft)

In this article, we investigate Reeb dynamics on bm-contact manifolds, previously introduced in [37], which are contact away from a hypersurface Zbut satisfy certain transversality conditions on Z. The study of these contact structures is motivated by that of contact manifolds with boundary. The search of periodic Reeb orbits on those manifolds thereby starts with a generalization of the well-known Weinstein conjecture. Contrary to the initial expectations, examples of compact bm-contact manifolds without periodic Reeb orbits outside Zare provided. Furthermore, we prove that in dimension 3, there are always infinitely many periodic orbits on the critical set if it is compact. We prove that traps for the bm-Reeb flow exist in any dimension. This investigation goes hand-in-hand with the Weinstein conjecture on non-compact manifolds having compact ends of convex, Postprint (published version)

In this article we introduce and analyze in detail singular contact structures, with an emphasis on $b^m$-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 $b^m$-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 those are related to smooth contact structures through a desingularization technique. The problem of existence of $b^m$-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 $b^m$-contact structure., In this article we introduce and analyze in detail singular contact structures, with an emphasis on $b^m$-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 $b^m$-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 those are related to smooth contact structures through a desingularization technique. The problem of existence of $b^m$-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 $b^m$-contact structure. 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 $b^{2k}$-contact manifold and $Z$ is its critical hypersurface. {As a consequence of the desingularization techniques in \cite{gmw1}, we prove the existence of folded contact forms on any almost contact manifold.}, Eva Miranda is supported by the Catalan Institution for Research and Advanced Studies via an ICREA AcademiaPrize 2016. C ́edric Oms is supported by an AFR-Ph.D. grant ofFNR - Luxembourg National Research Fund. Eva Mi-randa and C ́edric Oms are partially supported by the grants reference number MTM2015-69135-P (MINECO/FEDER)and reference number 2017SGR932 (AGAUR). Eva Miranda was supported by aChaire d’Excellenceof theFondationSciences Math ́ematiques de Pariswhen this project started and this work has been supported bya public grant overseenby the French National Research Agency (ANR) as part of the“Investissements d’Avenir”program (reference: ANR-10-LABX-0098). This material is based upon work supported by the National Science Foundation under Grant No.DMS-1440140 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, Califor-nia, during the Fall 2018 semester, Peer Reviewed, Preprint

In this article, we investigate Reeb dynamics onbm-contact manifolds, previously introduced in [MO], which are contact away from a hypersurface $Z$ but satisfy certain transversality conditions on $Z$., In this article, we investigate Reeb dynamics on $b^m$-contact manifolds, previously introduced in \cite{MO}, which are contact away from a hypersurface $Z$ but satisfy certain transversality conditions on $Z$. The study of these contact structures is motivated by that of contact manifolds with boundary. The search of periodic Reeb orbits on those manifolds thereby starts with a generalization of the well-known Weinstein conjecture. Contrary to the initial expectations, examples of compact $b^m$-contact manifolds without periodic Reeb orbits outside $Z$ are provided. Furthermore, we prove that in dimension $3$, there are always infinitely many periodic orbits on the critical set if it is compact. We prove that traps for the $b^m$-Reeb flow exist in any dimension. This investigation goes hand-in-hand with the Weinstein conjecture on non-compact manifolds having compact ends of convex type. In particular, we extend Hofer's arguments to open overtwisted contact manifolds that are $\R^+$-invariant in the open ends, obtaining as a corollary the existence of periodic $b^m$-Reeb orbits away from the critical set. The study of $b^m$-Reeb dynamics is motivated by well-known problems in fluid dynamics and celestial mechanics, where those geometric structures naturally appear. In particular, we prove that the dynamics on positive energy level-sets in the restricted planar circular three body problem are described by the Reeb vector field of a $b^3$-contact form that admits an infinite number of periodic orbits at the critical set., Eva Miranda is supported by the Catalan Institution for Research and Advanced Studies via an ICREA AcademiaPrize 2016. C ́edric Oms is supported by an AFR-Ph.D. grant of FNR - Luxembourg National Research Fund. Eva Mi-randa and C ́edric Oms are partially supported by the grants reference number MTM2015-69135-P (MINECO/FEDER)and reference number 2017SGR932 (AGAUR). Eva Miranda was supported by aChaire d’Excellenceof theFondationSciences Math ́ematiques de Pariswhen this project started and this work has been supported by a public grant overseenby the French National Research Agency (ANR) as part of the“Investissements d’Avenir”program (reference: ANR-10-LABX-0098). This material is based upon work supported by the National Science Foundation under Grant No.DMS-1440140 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, Califor-nia, during the Fall 2018 semester., Preprint

Motivated by Poincaré's orbits going to infinity in the (restricted) three-body problem (see \cite{poincare} and \cite{chenciner}), 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~\cite{CMP} of Etnyre--Ghrist's contact/Beltrami correspondence~\cite{EG}, and genericity results concerning eigenfunctions of the Laplacian established by Uhlenbeck \cite{uhlenbeck}. 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., This work is supported in part by the ICMAT–Severo Ochoa grant SEV-2015-0554 and the CSIC grant20205CEX001, Award-winning, Preprint

In this paper we analyze in detail a collection of motivating examples to consider bm-symplectic forms and folded-type symplectic structures. In particular, we provide models in Celestial Mechanics for every bm-symplectic structure. At the end of the paper, we introduce the odd-dimensional analogue to b-symplectic manifolds: b-contact manifolds., Peer Reviewed, Preprint

Morse homology studies the topology of smooth manifolds by examining the critical points of a real-valued function defined on the manifold, and connecting them with the negative gradient of the function. Rather surprisingly, the resulting homology is proved to be independent of the choice of the real-valued function and metric defining the negative gradient. This leads to a topological lower bound on the number of critical points. In the 1980s, the construction of Morse homology served as a prototype to define a homology spanned by $1$-periodic Hamiltonian diffeomorphisms on symplectic manifolds. The resulting homology, introduced by Andreas Floer, spectacularly revolutionized the area of symplectic topology and leaded to a proof of the famous Arnold conjecture. Floer theory still is the subject of a lot of active and exciting research and is nowadays an essential technique in symplectic topology.

In this paper we analyze in detail a collection of motivating examples to consider bm- symplectic forms and folded-type symplectic structures. In particular, we provide models in Celestial Mechanics for every bm-symplectic structure. At the end of the paper, we introduce the odd-dimensional analogue to b-symplectic manifolds: b-contact manifolds., Peer Reviewed, Preprint

En aquesta tesi estudiarem la relació entre les varietats simplèctiques tòriques i els sistemes integrables. Per això, presentem en primer lloc tots els preliminars necessaris de geometria simplèctica. Després presentem el teorema de Delzant i alguns exemples. Finalment, s'estudia i es presenta una demostració alternativa del teorema que relaciona les varietats amb els sistemes integrables: el teorema de Arnold-Liouville.