Is the derived set of an orbit differentiable?

Question

Let $M$ be a manifold and $f$ a diffeomorphism. Let us fixe a point $p$ and consider the orbit of $p$ under $f$, $f^n(p)$, $n \in {\bf Z}$. Let $N$ be the derived set of the orbit (the set of accumulation points), then has $N$ a differentiable structure? Is $N$ a immerged submanifold of $M$?