This is the first in a series of papers which develops a theory of coding for a class of \(\mathbb Z^2\) subshifts of finite types (SFTs) that is similar to the theory available for \(\mathbb Z\) mixing SFTs. The main result is the following Theorem:
For \(d\geq 2\), let \(X\) be a \(\mathbb Z^d\) subshift without periodic points and let \(Y\) be a \(\mathbb Z^d\) square mixing SFT containing a finite orbit. Suppose there is a homomorphism \(X\to Y\). Then, there exists an embedding \(X \hookrightarrow Y\) if and only if \(h(X)<h(Y)\). (It is also shown that for \(d=2\), a \(\mathbb Z^d\) square mixing SFT has finite orbits. Whether this is the case for \(d>2\) is an open problem.)
The author also announces the following result from his forthcoming paper:
If \(X\) is a \(\mathbb Z^2\) subshift without periodic points and if \(Y\) is a \(\mathbb Z^2\) square filling mixing SFT, then there exists a homomorphism \(X\to Y\). The two theorems together imply the following extension of Krieger’s Embedding Theorem:
Let \(X\) be a \(\mathbb Z^2\) subshift without periodic points and let \(Y\) be a \(\mathbb Z^2\) square filling mixing SFT. Then, there exists an embedding \(X \hookrightarrow Y\) if and only if \(h(X)<h(Y)\).