Strongly stable real infinitesimally symplectic mappings

Abstract

We prove that a mapA εsp(σ,R), the set of infinitesimally symplectic maps, is strongly stable if and only if its centralizerC(A) insp(σ,R) contains only semisimple elements. Using the theorem that everyB insp(σ,R) close toA is conjugate by a real symplectic map to an element ofC(A), we give a new proof of the openness of the set of strongly stable maps. Then we prove that the set of strongly stable maps is the interior of the set of all infinitesimally symplectic maps with purely imaginary or zero eigenvalues, and the connected components of this set are described. Finally, we give a new proof of the analytic conjugacy theorem for an analytic curve through a given strongly stable map.