- 00-297 Jorba A., Villanueva J.
- The fine geometry of the Cantor families of invariant tori
in Hamiltonian systems
(94K, PostScript, gzipped and uuencoded)
Jul 19, 00
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Abstract. This work focuses on the dynamics around a partially elliptic, lower
dimensional torus of a real analytic Hamiltonian system. More
concretely, we investigate the abundance of invariant tori in the
directions of the phase space corresponding to elliptic modes of the
torus. Under suitable (but generic) non-degeneracy and non-resonance
conditions, we show that there exists plenty of invariant tori in
these elliptic directions, and that these tori are organized in
manifolds that can be parametrized on suitable Cantor sets. These
manifolds can be seen as ``Cantor centre manifolds'', obtained as the
nonlinear continuation of any combination of elliptic linear modes of
the torus. Moreover, for each family, the density of the complementary
of the set filled up by these tori is exponentially small with respect
to the distance to the initial torus. These results are valid in the
limit cases when the initial torus is an equilibrium point or a
maximal dimensional torus. It is remarkable that, in the case in which
the initial torus is totally elliptic, we can derive Nekhoroshev-like
estimates for the diffusion time around the torus. Due to the use of
weaker non-resonance conditions, these results are an improvement on
previous results by the authors (Nonlinearity, 1997).
Talk given at the Third European Congress of Mathematics (Mini-symposium
on Symplectic and Contact Geometry and Hamiltonian Dynamics).
Barcelona, July 10th to 14th, 2000.