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Let ($M$, $\Omega$) be a smooth symplectic manifold and $f:M\rightarrow M$ bea symplectic diffeomorphism of class $C^l$ ($l\geq 3$). Let $N$ be a compactsubmanifold of $M$ which is boundaryless and normally hyperbolic for $f$. Wesuppose that $N$ is controllable and that its stable and unstable bundles aretrivial. We consider a $C^1$-submanifold $\d$ of $M$ whose dimension is equalto 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 thatfor 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 ofshadowing orbits for a finite family of invariant minimal sets (for which we donot assume any regularity) contained in a normally hyperbolic manifold andhaving heteroclinic connections. As a particular case, we recover classicalresults on the existence of diffusion orbits (Arnold's example).

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