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

Abstract

In 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.