Density of the set of symbolic dynamics with all ergodic measures supported on periodic orbits

Let $K$ be the Cantor set. We prove that arbitrarily close to a homeomorphism $T:K\to K$ there exists a homeomorphism $\tilde{T}:K\to K$ such that the $\alpha$-limit and the $\omega$-limit of every orbit is a periodic orbit. We also prove that arbitrarily close to an endomorphism $T:K\to K$ there exists an endomorphism $\tilde{T}:K\to K$ close to $T$ such that every orbit is finally periodic.