Borsuk-Ulam Theorems for Arbitrary S1 Actions and Applications

摘要

An S to the first power version of the Borsuk-Ulam Theorem is proved for a situation where Fix S to the first power may be nontrivial. The proof is accomplished with the aid of a new relative index theory. Applications are given to intersection theorems and the existence of multiple critica points is established for a class of functionals invariant under an S to the first power symmetry. Minimax arguments from the calculus of variations serve as an important tool in establishing the existence of nonlinear vibrations of discrete mechanical systems as modelled by Hamilton's equations. In these arguments one obtains the solutions of the differential equations as critical points of an associated Lagrangian by minimaxing the Lagrangian over appropriate classes of sets. Intersection theorems such as are proved in this paper play a crucial role in this process. In addition to obtaining some intersection theorems this report illustrates their use by proving an existence theorem for multiple critical points of the functional invarianet under an S to the first power symmetry group.