Fundamental Open Problems on Area Preserving Mappings

In this note we present a number of open problems and conjectures about area preserving mappings of the plane. In specific instances we will illustrate these problems via a number of well known examples: the standard family and the conservative Henon family. We try to further develop a connection with renormalization and holomorphic dynamics. We believe the time is ripe for renewed vigor. Poincare, more or less single handledly, laid bare the astonishing complexity of the dynamics of Hamiltonian systems, and saw the limits of the computational analytical point of view: to 'compute' the dynamics using for instance power series expansions. In the process of analyzing the dynamics by a geometric decomposition of phase space Poincare discovered many of the fundamental notions of dynamics known today: periodic points, their eigenvalues, ellipticity and hyperbolicity, invariant manifolds, heteroclinic and homoclinic intersections, normal forms, and genericity.