Rabinowitz Floer homology for tentacular Hamiltonians

Pasquotto, Federica; Vandervorst, Robert; Wiśniewska, Jagna

doi:10.48550/arxiv.1810.05432

Cited by 3 publications

(11 citation statements)

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“…The homology of the chain complex (CF * (H, f ), ∂) is called the Rabinowitz Floer homology of H and is denoted by RF H * (H). For the Hamiltonians H that we consider in Theorem 1.1, the fact that RF H * (H) is well defined and independent of the auxiliary data used to construct it is proved in [32,31]. One can also play the same game with the Hamiltonian H 0 on T * R k , thus yielding another Rabinowitz Floer homology RF H * (H 0 ).…”

Section: Introductionmentioning

confidence: 94%

“…Below we will outline the construction of Rabinowitz Floer homology groups for Hamiltonians H ∈ H. In the case H satisfies (c), this reduces to the original definition presented by Cieliebak and Frauenfelder [14], only specialised to T * R m . Meanwhile for strongly tentacular H, the construction 6 comes from [31].…”

Section: Sign Conventionsmentioning

confidence: 99%

“…Compactness up to breaking implies that ∂ 2 = 0, and a continuation argument tells us that the resulting Rabinowitz Floer homology RF H(H) is independent of the auxiliary data used to define it. We refer the reader to [14] (when H satisfies (c)) and [31] (when H is strongly tentacular) for details.…”

Section: Sign Conventionsmentioning

confidence: 99%

“…We say that H satisfies property (PO) if for any fixed action window, all non-degenerate periodic orbits are contained in a compact subset of M . Moreover, we say that property (PO) is uniformly continuous at H if there exists an open neighbourhood O(H) of 0 in C ∞ c (M ) and an exhaustion of M by compact sets {K n } n∈N , such that for every n ∈ N and every h [31,Lem. 8.4] every strongly tentacular Hamiltonian satisfies the axiom of uniform continuity of (PO).…”

Section: Invariance Of the Positive Rabinowitz Floer Homologymentioning

confidence: 99%

“…The non-compact case is rather less tractable. In [32,31] the class of non-compact hypersurfaces for which the Rabinowitz Floer homology could be defined was extended, and a rather general invariance result was proved. However no explicit computations were presented.…”

Section: Introductionmentioning

confidence: 99%

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Computing the Rabinowitz Floer homology of tentacular hyperboloids

Fauck¹,

Merry²,

Wiśniewska³

2020

Preprint

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We compute the Rabinowitz Floer homology for a class of noncompact hyperboloids Σ ≃ S n+k−1 × R n−k . Using an embedding of a compact sphere Σ0 ≃ S 2k−1 into the hypersurface Σ, we construct a chain map from the Floer complex of Σ to the Floer complex of Σ0. In contrast to the compact case, the Rabinowitz Floer homology groups of Σ are both non-zero and not equal to its singular homology. As a consequence, we deduce that the Weinstein Conjecture holds for any strongly tentacular deformation of such a hyperboloid.

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Section: Introductionmentioning

confidence: 94%

Section: Sign Conventionsmentioning

confidence: 99%

Section: Sign Conventionsmentioning

confidence: 99%

Section: Invariance Of the Positive Rabinowitz Floer Homologymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Computing the Rabinowitz Floer homology of tentacular hyperboloids

Fauck¹,

Merry²,

Wiśniewska³

2020

Preprint

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Helium and Hamiltonian delay equations

Frauenfelder¹

2020

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Global Hamiltonian dynamics on singular symplectic manifolds

Oms

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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.

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Rabinowitz Floer homology for tentacular Hamiltonians

Cited by 3 publications

References 12 publications

Computing the Rabinowitz Floer homology of tentacular hyperboloids

Computing the Rabinowitz Floer homology of tentacular hyperboloids

Helium and Hamiltonian delay equations

Global Hamiltonian dynamics on singular symplectic manifolds

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