An inclination lemma for normally hyperbolic manifolds with an application to diffusion

TL;DR

This paper proves a new inclination lemma for normally hyperbolic manifolds in symplectic dynamics, demonstrating how certain submanifolds approximate unstable manifolds and applying this to establish the existence of shadowing and diffusion orbits.

Contribution

It introduces a novel inclination lemma for normally hyperbolic manifolds with trivial bundles and applies it to prove the existence of shadowing and diffusion orbits in symplectic systems.

Findings

Existence of points with orbits close to unstable manifolds

Shadowing orbits for invariant minimal sets with heteroclinic connections

Recovery of classical diffusion results in Hamiltonian dynamics

Abstract

Let ($M$, $\Omega$) be a smooth symplectic manifold and $f:M\rightarrow M$ be a symplectic diffeomorphism of class $C^l$ ($l\geq 3$). Let $N$ be a compact submanifold of $M$ which is boundaryless and normally hyperbolic for $f$. We suppose that $N$ is controllable and that its stable and unstable bundles are trivial. We consider a $C^1$-submanifold $\d$ of $M$ whose dimension is equal to the dimension of a fiber of the unstable bundle of $T_NM$. We suppose that $\d$ transversely intersects the stable manifold of $N$. Then, we prove that for all $\varepsilon>0$, and for $n$ $\in$ $\N$ large enough, there exists $x_n$ $\in$ $N$ such that $f^n(\d)$ is $\varepsilon$-close, in the $C^1$ topology, to the strongly unstable manifold of $x_n$. As an application of this $\lambda$-lemma, we prove the existence of shadowing orbits for a finite family of invariant minimal sets (for which we do not assume any regularity) contained in a normally hyperbolic manifold and having heteroclinic connections. As a particular case, we recover classical results on the existence of diffusion orbits (Arnold's example).