# On Krieger's Embedding Theorem

This is Theorem 10.1.1 of Lind & Marcus's book, An Introduction to Symbolic Dynamics and Coding. They say that is "straightfordward" to go from

Let $$X$$ a shift of finite type and $$Y$$ a mixing shift of finite type such that $$\text{Per}(X)\hookrightarrow\text{Per}(Y)$$ and $$h(X). Then, $$X\hookrightarrow Y$$.

to

Let $$X$$ and $$Y$$ irreducible shift of finite type such that $$\text{Per}(X)\hookrightarrow\text{Per}(Y)$$ and $$h(X). Then, $$X\hookrightarrow Y$$.

How we can drop the mixing hypothesis on $$Y$$? I have thought in this all new year!

the difference between the two statements is rather subtle. Of course proving the result with $$X$$ any SFT is more general than assuming $$X$$ to be irreducible, so there is nothing to do there. For $$Y$$, going from irreducible to mixing seems to be a stronger condition, however the structure threoy of (one-dimensional) SFTs tells us that a non-mixing irreducible SFT is merely a finite union of mixing ones, all conjugate to each other. If you have not seen this, take a look at Chapter 4.5 of Lind-Marcus. Hence the conditions of the embedding theorem still hold and we might look at one of the cyclic components (i.e. a mixing SFT) of $$Y$$ instead.