The equivariant spectral flow and bifurcation of periodic solutions of Hamiltonian systems

Abstract

We define a spectral flow for paths of selfadjoint Fredholm operators that are equivariant under the orthogonal action of a compact Lie group as an element of the representation ring of the latter. This G-equivariant spectral flow shares all common properties of the integer valued classical spectral flow, and it can be non-trivial even if the classical spectral flow vanishes. Our main theorem uses the G-equivariant spectral flow to study bifurcation of periodic solutions for autonomous Hamiltonian systems with symmetries.