  Let $K$ be the Cantor set. We prove that arbitrarily close to a homeomorphism
$T:K\rightarrow K$ there exists a homeomorphism $\widetilde T:K\rightarrow K$
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
$T:K\rightarrow K$ there exists an endomorphism $\widetilde T:K\rightarrow K$
close to $T$ such that every orbit is finally periodic.
