Shadowing Lemmas in Normally Hyperbolic Invariant Manifolds and Applications to the Arnold Diffusion Problem
speaker: Marian Gidea (Yeshiva University New York )abstract: We describe two shadowing lemma type of results for normally hyperbolic invariant manifolds, as follows. Given a normally hyperbolic invariant manifold whose stable and unstable manifolds intersect transversally along a transverse homoclinic manifold, we associate two dynamics. One is the scattering map, which accounts for the `outer dynamics' along homoclinic orbits, and the other is the restriction to the normally hyperbolic invariant manifold, which accounts for the `inner dynamics'. The main results can be summarized as follows: (i) For every pseudo-orbit generated by alternatively applying the scattering map and the inner dynamics for sufficiently long time, there exists a true orbit of the system near that pseudo-orbit; (ii) Assuming that almost every point in the normally hyperbolic invariant manifold is recurrent relative to the inner dynamics, then for every pseudo-orbit obtained by successively applying the scattering map, there exists a true orbit near that pseudo-orbit. These results remain true if one considers several transverse homoclinic manifolds rather than a single one, and the appropriate scattering maps.
As an application, we describe a qualitative result on the existence of diffusing orbits in a priori unstable Hamiltonian systems, under generic conditions on the perturbation that are verifiable in concrete systems, and under some mild conditions on the unperturbed system. Most notably, we do not require any conditions on the inner dynamics.
This is based on joint work with R. de la Llave and T. Seara.
timetable:
Wed 7 May, 9:00 - 10:00, Aula Dini
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