Birkhoff-Lewis Fixed Point Theorem and a Conjecture of V. I. Arnold

摘要

Periodic solutions of Hamiltonian systems are also critical points of a function on the loop space of the underlying phase space. If this functional is bounded below, Morse's theory of critical points applies and he made such an application to the problem of closed geodesics. In the present problem (and in many more which arise in physics) the functional is not bounded below and in fact tends to + infinity and - infinity on (different) infinite dimensional sets. Understanding such 'infinitely indefinite' functionals is basic for mathematical physics. The fundamental work of P. Rabinowitz set the tone for overcoming this difficulty. It's modification here solves (the simplest version) of one of the key problems of symplectic geometry. (Author)