Emerging Topics Working Group

Abstract: In this talk we present a general shadowing result for normally hyperbolic invariant manifolds. The result does not use the existence of invariant objects like tori inside the manifold and works in very general settings. We apply this result to establish the existence of diffusing orbits in a large class of nearly integrable Hamiltonian systems. Our approach relies on successive applications of the so called `scattering map' along homoclinic orbits to a normally hyperbolic invariant manifold. The main idea is that we can closely follow any path of the scattering map. This gives the existence of diffusing orbits. The method applies to perturbed integrable Hamiltonians of arbitrary degrees of freedom (not necessarily convex) which present some hyperbolicity without any assumption about the inner dynamics. Joint work with Marian Gidea and Rafael de la Llave.