An embedding theorem for multidimensional subshifts

AbstractKrieger’s embedding theorem provides necessary and sufficient conditions for an arbitrary subshift to embed in a given topologically mixing $\mathbb {Z}$ -subshift of finite type. For certain families of $\mathbb {Z}^d$ -subshifts of finite type, Lightwood characterized the aperiodic subsystems. In the current paper, we prove a new embedding theorem for a class of subshifts of finite type over any countable abelian group. Our theorem provides necessary and sufficient conditions for an arbitrary subshift X to embed inside a given subshift of finite type Y that satisfies a certain natural condition. For the particular case of $\mathbb {Z}$ -subshifts, our new theorem coincides with Krieger’s theorem. Our result gives the first complete characterization of the subsystems of the multidimensional full shift $Y= \{0,1\}^{\mathbb {Z}^d}$ . The natural condition on the target subshift Y, introduced explicitly for the first time in the current paper, is called the map extension property. It was introduced implicitly by Mike Boyle in the early 1980s for $\mathbb {Z}$ -subshifts and is closely related to the notion of an absolute retract, introduced by Borsuk in the 1930s. A $\mathbb {Z}$ -subshift has the map extension property if and only if it is a topologically mixing subshift of finite type. We show that various natural examples of $\mathbb {Z}^d$ subshifts of finite type satisfy the map extension property, and hence our embedding theorem applies for them. These include any subshift of finite type with a safe symbol and the k-colorings of $\mathbb {Z}^d$ with $k \ge 2d+1$ . We also establish a new theorem regarding lower entropy factors of multidimensional subshifts that extends Boyle’s lower entropy factor theorem from the one-dimensional case.