Variations on Krieger's Embedding Theorem and a Theorem of Boyle

Krieger's celebrated embedding theorem gives necessary and sufficient conditions, in terms of periodic points, for proper embedding of a subshift into a mixing shift of finite type (SFT). The result does not generalize to mixing sofic shifts. Boyle introduced the notion of a receptive periodic point, which is the main obstruction. He also proved that a condition involving receptive periodic points is sufficient for proper embedding of a subshift into a mixing sofic shift. We introduce a stronger notion of embedding, called factorizable embedding, which requires the embedding to factor through a mixing SFT. We show that Boyle's sufficient condition for proper embedding into a mixing sofic shift is necessary and sufficient for a factorizable embedding into a mixing sofic shift. We also give a new characterization of receptive periodic points, and we give a characterization of mixing for irreducible sofic shifts in terms of receptive periodic points. This is joint work with Tom Meyerovitch, Klaus Thomsen and Chengyu Wu.