Subshift attractors of cellular automata

A subshift attractor is a two-sided subshift which is an attractor of a cellular automaton. We prove that each subshift attractor is chain-mixing, contains a configuration which is both F-periodic and sigma-periodic and the complement of its language is recursively enumerable. We prove that a subshift of finite type is an attractor of a cellular automaton iff it is mixing. We identify a class of chain-mixing sofic subshifts which are not subshift attractors. We construct a cellular automaton whose maximal attractor is a non-sofic mixing subshift, answering a question raised in Maass (1995 Ergod. Theory Dyn. Syst. 15 663-84). We show that a cellular automaton is surjective on its small quasi-attractor which is the non-empty intersection of all subshift attractors of all F-q sigma(p), where q > 0 and p is an element of Z.