Diffusion along chains of normally hyperbolic cylinders

Arnold diffusion and Mather theory - Ke Zhang

Marian Gidea
Yeshiva University
April 11, 2018
Abstract: We consider a geometric framework that can be applied to prove the existence of drifting orbits in the Arnold diffusion problem. The main geometric objects that we consider are 3-dimensional normally hyperbolic invariant cylinders with boundary, which admit well-defined stable and unstable manifolds. These enable us to define chains of cylinders i.e., finite, ordered families of cylinders in which each cylinder admits homoclinic connections, and any two consecutive cylinders admit heteroclinic connections. We show the existence of orbits drifting along such chains, under precise conditions on the dynamics on the cylinders, and on their homoclinic and heteroclinic connections.
Our framework applies to both the a priori stable setting, once the preliminary geometric reductions are preformed, and to the a priori unstable setting, rather directly. This is joint work with J.-P. Marco.