Regular maps with the specification property

Let $f$ be a $C^1$-regular map of a closed $C^{\infty}$ manifold $M$ and $\Lambda$ be a locally maximal closed invariant set of $f$. We show that $f|_{\Lambda}$ satisfies the $C^1$-stable specification property if and only if $\Lambda$ is a hyperbolic elementary set. We also prove that there exists a residual subset $\mathcal{R}$ in the space of $C^1$-regular maps endowed with the $C^1$-topology such that for $f \in \mathcal{R}$, $f|_{\Lambda}$ satisfies the specification property if and only if $\Lambda$ is a hyperbolic elementary set.