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Spectral invariant of Floer homology and its application to Hill's lunar problem

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dc.contributor.author이준영-
dc.date.accessioned2017-07-14T00:42:44Z-
dc.date.available2017-07-14T00:42:44Z-
dc.date.issued2016-08-
dc.identifier.other000000136999-
dc.identifier.urihttps://hdl.handle.net/10371/121318-
dc.description학위논문 (박사)-- 서울대학교 대학원 : 수리과학부, 2016. 8. Otto van Koert.-
dc.description.abstractIn this thesis, we reinterpret spectral invariants in symplectic homology and wrapped Floer homology as symplectic capacities for fiberwise star-shaped domains in a cotangent bundle of a closed orientable manifold. We compute the spectral invariants of various homology classes in symplectic homology and wrapped Floer homology for the fiberwise star-shaped domains defined by the rotating Kepler problem. Moreover we prove inclusions among the fiberwise star-shaped domains defined by the rotating Kepler problem and Hill's lunar problem. Finally if we combine computations of spectral invariants and result of inclusions, then we obtain estimates for spectral invariants in the symplectic homology and the wrapped Floer homology of Hill's lunar problem using monotonicity of spectral invariants. As a result, using spectrality of spectral invariants, these estimates for spectral invariants of Hill's lunar problem give us estimates of the action values of periodic orbits, symmetric periodic orbits and doubly symmetric orbits in Hill's lunar problem. As a Corollary, we can obtain systole bounds for the regularized Hill's lunar problem: For $c > c_H^0$, there is at least one periodic Reeb orbit whose action is less than $\pi$ on $(\Sigma_H^c, \lambda_{can})$. Moreover, we can say the same result for symmetric periodic Reeb orbits and for doubly symmetric periodic Reeb orbits. Furthermore, we obtain a sequence of intervals which insure the existence of a (symmetric) periodic orbit whose action lies on each of the intervals.-
dc.description.tableofcontentsChapter 1. Introduction 1

Chapter 2. Preliminaries 10
2.1. Hamiltonian dynamics and Symplectic geometry 10
2.2. Integrals and Completely Integrable Systems 20
2.3. Geodesic Problems 24
2.4. Contact geometry and Liouville domain 29

Chapter 3. The restricted three body problem and its limit problems 34
3.1. The restricted three body problem 35
3.2. The rotating Kepler problem 40
3.3. Hill's lunar problem 41

Chapter 4. Moser regularization 44
4.1. Kepler problem and Moser regularization 44
4.2. Fiberwise convexity 46
4.3. Fiberwise star-shapedness 49

Chapter 5. Floer homology 52
5.1. Symplectic homology of Liouville domain 52
5.2. Wrapped Floer homology of Liouville domain 69

Chapter 6. Spectral invariants for fiberwise star-shaped domains in cotangent bundles 83
6.1. Spectral invariant in symplectic homology 83
6.2. Spectral invariant in wrapped Floer homology 93

Chapter 7. Application to Hill's lunar problem 96
7.1. The Maslov indices of the rotating Kepler problem 97
7.2. Spectrum of the rotating Kepler problem 109
7.3. Computation of spectral invariant for the rotating Kepler problem 114
7.4. Inclusions between the rotating Kepler problem and Hill's lunar problem 120
7.5. Estimates for spectral invariants of Hill's lunar problem 127

Appendix: Maslov index 136
Appendix: Maslov indices for paths of Lagrangian subspaces 136
Appendix: Maslov indices on cotangent bundles 140

Abstract (in Korean) 149
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dc.formatapplication/pdf-
dc.format.extent3750108 bytes-
dc.format.mediumapplication/pdf-
dc.language.isoen-
dc.publisher서울대학교 대학원-
dc.subjectHamiltonian Dynamics-
dc.subjectThe Rotating Kepler Problem-
dc.subjectHill's Lunar Problem-
dc.subjectFloer Homology-
dc.subjectSpectral Invariant-
dc.subjectFiberwise Convexity-
dc.subject.ddc510-
dc.titleSpectral invariant of Floer homology and its application to Hill's lunar problem-
dc.typeThesis-
dc.description.degreeDoctor-
dc.citation.pages157-
dc.contributor.affiliation자연과학대학 수리과학부-
dc.date.awarded2016-08-
Appears in Collections:
College of Natural Sciences (자연과학대학)Dept. of Mathematical Sciences (수리과학부)Theses (Ph.D. / Sc.D._수리과학부)
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