Topological shadowing methods in arnold diffusion: weak torsion and multiple time scales

Abstract

Consider a symplectic map which possesses a normally hyperbolic in- variant manifold of any even dimension with transverse homoclinic chan- nels. We develop a topological shadowing argument to prove the existence of Arnold di usion along the invariant manifold, shadowing some itera- tions of the inner dynamics carried by the invariant manifold and the outer dynamics induced by the stable and unstable foliations. In doing so, we generalise an idea of Gidea and de la Llave in [26], based on the method of correctly aligned windows and a so-called transversality-torsion argument. Our proof permits that the dynamics on the invariant mani- fold satisfy only a non-uniform twist condition, and, most importantly for applications, that the splitting of separatrices be small in certain direc- tions and thus the associated drift in actions very slow; di usion occurs in the directions of the manifold having non-small splitting. Furthermore we provide estimates for the di usion time.