The Closed Characteristics On P-symmetric Compact Convex Hypersurfaces In R²ⁿ

In this thesis,we study the closed characteristics on P-symmetric compact convex hypersurfaces in R2n,where P is a symplectic matrix,P-I2n is invertible and Pm=I2n for some m>1.There are three aspects:the Multiplicity,Stability and Resonance identity(i.e.the relationship between the closed characteristics).For the multiplicity problem,we apply the Maslov-type index theory and Maslov index theory to prove that if P satisfies Pm=I2n and similar to R(?)?[n/2]?R(-?)?n-[n/2],then there is at least[3n/4]+1 many closed characteristics on any P-symmetric compact convex hypersurface.For the stability problem,we use the variational method and Morse theory to prove that for a certain class of symplectic matrices P,there exist an elliptic P-symmrtric closed characteristic on any P-symmetric compact convex hyper-surface.For the relations between the closed characteristics,with the help of the finite dimensional reduction,equivariant Morse theory and P-index iteration theory,it is proved that a resonance identity of all P-symmetric prime closed charac-teristics can be established when the number of all P-symmetric prime closed characteristics on P-symmetric compact convex hypersurfaces is finite and the minimal period of critical modules exists for each of them.In particular,if a P-symmetric prime closed characteristics satisfy P boundary condition and strongly non-degenerate,the minimal period of critical modules exists.