  We study the density of periodic configurations for shift spaces defined on
(the Cayley graph of) a finitely generated group. We prove that in the case of
a full shift on a residually finite group and in that of a group shift space on
an abelian group, the periodic configurations are dense. In the one-dimensional
case we prove the density for irreducible sofic shifts. In connection with this
we study the surjunctivity of cellular automata and local selfmappings. Some
related decision problems for shift spaces of finite type are also
investigated.
