Families of whiskered tori for a-priori stable/unstable Hamiltonian systems and construction of unstable orbits

We give a detailed statement of a KAM theorem about the conservation of partially hyperbolic tori on a fixed energy level for an analytic Hamiltonian H(I,φ, p, q) = h(I, pq;μ) + μf(I, φ, p, q;μ), where φ is a (d − 1)−dimensional angle, I is in a domain of R, p and q are real in a neighborhood 0, and μ is a small parameter. We show that invariant whiskered tori covering a large measure exist for sufficiently small perturbations. The associated stable and unstable manifolds also cover a large measure. Moreover, we show that there is a geometric organization to these tori. Roughly, the whiskered tori we construct are organized in smooth families, indexed by a Cantor parameter. The whole set of tori as well as their stable and unstable manifolds is smoothly interpolated. In particular, we emphasize the following items: sharp estimates on the relative measure of the surviving tori on the energy level, analyticity properties, including dependence upon parameters, geometric structures. We apply these results to both “a-priori unstable” and “a-priori stable” systems. We also show how to use the information obtained in the KAM Theorem we prove to construct unstable orbits.