Closed Characteristics On Hamiltonian Energy Surfaces

This thesis is about the periodic orbits on Hamiltonian energy surfaces （we call them closedcharacteristics）, it consists of four parts.In the first part, we study relationship between the stability and the growth of the number ofclosed characteristics on compact convex hypersurfaces in R²ⁿ, and also the stability of closedcharacteristics on compact convex hypersurfaces in R²ⁿunder some pinching conditions.In the second part, we study the stability of symmetric closed characteristics on symmetriccompact convex hypersurfaces in R²ⁿunder a pinching condition，and we establish a newresonance identity for symmetric closed characteristics on symmetric compact convex hyper-surface Σ in R²ⁿprovided that there are finitely many geometrically distinct symmetric closedcharacteristics. As its applications, some interesting results about the stability and multiplicityof symmetric closed characteristics are obtained, and also we prove that if Σ is C∞-generic, itcarries infinitely many symmetric closed characteristics.In the third part, we establish two new resonance identities for closed characteristics onevery compact star-shaped hypersurface Σ in R²ⁿwhen the number of geometrically distinctclosed characteristics on Σ is finite. These identities extend those of C. Viterbo on the non-degenerate case in1989, and that of W. Wang, X. Hu and Y. Long on convex case in2007.Then using these identities and the index iteration theory, we study the multiplicity of closedcharacteristics on compact star-shaped hypersurfaces.In the fourth part, let P=diag(-I_n-κ, I_κ,-I_n-κ, I_κ) for some integer κ∈[0,n], we provethat for any compact convex hypersurfaces Σ in R²ⁿwith n≥2there exist at least two geometri-cally distinct P-invariant closed characteristics on Σ, provided that Σ is P-symmetric, i.e., x∈Σimplies Px∈Σ. This work is shown to extend and unify several earlier works on this subject.