Iteration Formulae for Brake Orbit and Index Inequalities for Real Pseudoholomorphic Curves

Zhou, Beijia

doi:10.48550/arxiv.2011.07958

Cited by 1 publication

(1 citation statement)

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“…n = 1. This case has also been studied in [17,Appendix B], where it plays an important role when trying to define a real version of ECH. The identification…”

Section: The Symplectic Group Symmetries and Git Quotientsmentioning

confidence: 99%

On GIT quotients of the symplectic group, stability and bifurcations of symmetric orbits

Frauenfelder¹,

Moreno²

2021

Preprint

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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. CONTENTS 1. Introduction 2. Geometric and dynamical setup 3. The symplectic group, symmetries, and GIT quotients 4. The characteristic polynomial 5. Normal forms 6. Bifurcations and stability Appendix A. The GIT quotient Appendix B. Krein theory and strong stability for Hamiltonian systems References

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“…n = 1. This case has also been studied in [17,Appendix B], where it plays an important role when trying to define a real version of ECH. The identification…”

Section: The Symplectic Group Symmetries and Git Quotientsmentioning

confidence: 99%

On GIT quotients of the symplectic group, stability and bifurcations of symmetric orbits

Frauenfelder¹,

Moreno²

2021

Preprint

View full text Add to dashboard Cite

show abstract

Iteration Formulae for Brake Orbit and Index Inequalities for Real Pseudoholomorphic Curves

Abstract: I give precise iteration formulae for brake orbit in dimension 3 and use these formulae to get some index inequalities for moduli spaces of Real pseudoholomorphic Curves, which are important to establish Real embedded contact homology and Real cylindrical contact homology in dimension 3.

Cited by 1 publication

References 21 publications

On GIT quotients of the symplectic group, stability and bifurcations of symmetric orbits

On GIT quotients of the symplectic group, stability and bifurcations of symmetric orbits

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