Title: Diffusion along chains of normally hyperbolic cylinders,
Abstract: The present paper is part of a series of articles dedicated to the existence
of Arnold diffusion for cusp-residual perturbations of Tonelli Hamiltonians on
$\mathbb{A}^3$. Our goal here is to construct an abstract geometric framework
that can be used to prove the existence of diffusing orbits in the so-called a
priori stable setting, once the preliminary geometric reductions are preformed.
Our framework also applies, rather directly, to the a priori unstable setting.
The main geometric objects of interest are $3$-dimensional normally
hyperbolic invariant cylinders with boundary, which in particular admit
well-defined stable and unstable manifolds. These enable us to define, in our
setting, chains of cylinders, i.e., finite, ordered families of cylinders in
which each cylinder admits homoclinic connections, and any two consecutive
elements in the family admit heteroclinic connections.
Our main result is the existence of diffusing orbits drifting along such
chains, under precise conditions on the dynamics on the cylinders, and on their
homoclinic and heteroclinic structure.