An embedding theorem for multidimensional subshifts

Abstract

Krieger'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 some $\mathbb{Z}^d$-subshifts of finite type, Lightwood characterized the \emph{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 main 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 condition. For the particular case of $\mathbb{Z}$-subshifts, our new theorem coincides with Krieger's theorem. In particular, our result gives the first complete characterization of the subsystems of the multidimensional full shift $Y= A^{\mathbb{Z}^d}$. 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 1980's for $\mathbb{Z}$-subshifts, and is closely related to the notion of an absolute retract, introduced by Borsuk in the 1930's. A $\mathbb{Z}$-subshift has the map extension property if and only if it is a topologically mixing subshift of finite type. Over abelian groups, a subshift has the map extension property if and only if it is a contractible SFT as shown in work of Poirier and Salo. 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.