MPEJ Volume 6, No.2, 18 pp.
Received Nov 19 1999, Revised Feb 10 2000, Accepted Feb 15 2000
E. Valdinoci
Families of whiskered tori for a-priori stable/unstable
Hamiltonian systems and construction of unstable orbits
ABSTRACT:
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 $H(I,\f,p,q)=h(I,pq;\m)+\m f(I,\f,p,q;\m)$,
where $\f$ is a $({d}-1)-$dimensional angle, $I$ is in a domain of
$\RR^{{d}-1}$, $p$ and $q$ are real in a neighborhood $0$, and $\m$
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.