Towards a global study of Area Preserving Maps.
Examples, Models and Applications

C. Sim\'o, A. Vieiro
Dpt. Matem\`atica Aplicada i An\`alisi,
Universitat de Barcelona,
Gran Via, 585, 08007, Barcelona, Spain
carles@maia.ub.es, vieiro@maia.ub.es

We present a phenomenological study of Area Preserving Maps, with 
emphasis on simple models. Paradigmatic models like the standard map,
the separatrix map and a new model, the biseparatrix map, are useful to
undertand the dynamics. 

These paradigms can be analysed in a reasonable way. They allow to
explain the main features of the dynamics. As test example the
conservative H\'enon map is used.

Main points of interest are the escape of points from a given region 
when they are not surrounded by invariant rotational curves and the
measure of the set on non-escaping points. Different rates of escape
have been found. 

The existence of regions where the dynamics is close to a diffusion,
separated by cantori which are difficult to cross, having noble numbers
as rotation number, are illustrated.

Applications are also made to the unfolding of the Hopf-saddle-node
bifurcations for 3D vector fiels, to the stability regions around
triangular points in the RTBP as a function of the mass parameter,
and to the confiniment of outer cometary motions in the sun-jupiter
system as a function of the Jacobi constant. 

The talk is closed by several open questions.