Diffusing orbits along chains of cylinders

Abstract

<p style='text-indent:20px;'>We develop a geometric mechanism to prove the existence of orbits that drift along a prescribed sequence of cylinders, under some general conditions on the dynamics. This mechanism can be used to prove the existence of Arnold diffusion for large families of perturbations of Tonelli Hamiltonians on <inline-formula><tex-math id="M1">\begin{document}$ {{\mathbb A}}^3 $\end{document}</tex-math></inline-formula>. Our approach can also be applied to more general Hamiltonians that are not necessarily convex.</p><p style='text-indent:20px;'>The main geometric objects in our framework are <inline-formula><tex-math id="M2">\begin{document}$ 3 $\end{document}</tex-math></inline-formula>–dimensional invariant cylinders with boundary (not necessarily hyperbolic), which are assumed to admit center-stable and center-unstable manifolds. These enable us to define <i>chains of cylinders</i>, i.e., finite, ordered families of cylinders where each cylinder admits homoclinic connections, and any two consecutive cylinders in the chain admit heteroclinic connections.</p><p style='text-indent:20px;'>Our main result is on the existence of diffusing orbits which drift along such chains of cylinders, under precise conditions on the dynamics on the cylinders – i.e., the existence of Poincaré sections with the return maps satisfying a <i>tilt condition</i> – and on the geometric properties of the intersections of the center-stable and center-unstable manifolds of the cylinders – i.e., certain compatibility conditions between the <i>tilt map</i> and the homoclinic maps associated to its essential invariant circles.</p><p style='text-indent:20px;'>We give two proofs of our result, a very short and abstract one, and a more constructive one, aimed at possible applications to concrete systems.</p>