Chain mixing endomorphisms are approximated by subshifts on the Cantor set

Let f be a chain mixing continuous onto mapping from the Cantor set onto itself. Let g be a homeomorphism on the Cantor set that is topologically conjugate to a subshift. Then, homeomorphisms that are topologically conjugate to g approximate f in the topology of uniform convergence if a trivial necessary condition on the periodic points holds. In particular, if f is a chain mixing continuous onto mapping from the Cantor set onto itself with a fixed point, then homeomorphisms on the Cantor set that are topologically conjugate to a subshift approximate f in the topology of uniform convergence. In addition, homeomorphisms on the Cantor set that are topologically conjugate to a subshift without periodic points approximate any chain mixing continuous onto mappings from the Cantor set onto itself. In particular, let f be a homeomorphism on the Cantor set that is topologically conjugate to a full shift. Let g be a homeomorphism on the Cantor set that is topologically conjugate to a subshift. Then, a sequence of homeomorphisms that is topologically conjugate to g approximates f.