Symplectic invariants and Hamiltonian dynamics

Abstract

The discoveries of the last decades have opened new perspectives forthe old field of Hamiltonian systems and led to the creation of anew field: symplectic topology. Surprising rigidity phenomena demonstratethat the nature of symplectic mappings is very different from thatof volume preserving mappings. This raises new questions, many ofthem still unanswered. On the other hand, analysis of an old variationalprinciple in classical mechanics has established global periodicphenomena in Hamiltonian systems. As it turns out, these seeminglydifferent phenomena are mysteriously related. One of the links isa class of symplectic invariants, called symplectic capacities. Theseinvariants are the main theme of this book, which includes such topicsas basic symplectic geometry, symplectic capacities and rigidity,periodic orbits for Hamiltonian systems and the action principle,a bi-invariant metric on the symplectic diffeomorphism group andits geometry, symplectic fixed point theory, the Arnold conjecturesand first order elliptic systems, and finally a survey on Floer homologyand symplectic homology.The exposition is self-contained and addressed to researchers andstudents from the graduate level onwards.