Let $M$ be a compact smooth manifold and $N$ be a compact smooth submanifold of $M$. The usual transversality theorem claims that for a generic diffeomorphism $f$ of $M$, the submanifolds $N$ and $f(N)$ are transverse.
I am interested in another point of view. The diffeomorphism $f$ is now fixed and I wonder now if one can perturb smoothly the submanifold $N$ so that $N$ and $f(N)$ are transverse. Of course it requires some condition on $f$ as trivially it cannot be done for the identity map. Assume maybe first $f$ has no periodic points in $M$.