Topological conjugacy on the complement of the periodic points

We will say that two subshifts are essentially conjugate if they are topologically conjugate on the complement of their periodic points. In 1990, Susan Williams presented an example of a sofic shift that is not topologically conjugate to a renewal system. We show that the example of Williams is essentially conjugate to a renewal system. We also present an example of a renewal system that is essentially conjugate to a shift of finite type but not topologically conjugate to a shift of finite type. Finally, we prove that all renewal systems that meet a certain technical condition are essentially conjugate to a shift of finite type.