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

Abstract

Let ($M$, $\Omega$) be a smooth symplectic manifold and $f:M\to M$ be a symplectic diffeomorphism of class $C^l$ ($l\ge 3$). 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 $T_{N}M$. We suppose that $\d$ transversely intersects the stable manifold of $N$. Then, we prove that for all $\epsilon >0$, and for $n$ $\in$ $\mathbb{N}$ large enough, there exists $x_n$ $\in$ $N$ such that $f^n(\d)$ is $\epsilon$-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).

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