# Sliding block code on irreducible sofic shift

I was looking at the following exercise from Lind/Marcus book An Introduction to Symbolic Dynamics and Coding that I cannot solve. Can someone give me a hint?

Find an example of a pair of irreducible sofic shifts $$X,Y$$ and a sliding block code $$f:X\rightarrow Y$$ such that $$f$$ is not one-to-one, but the restriction of f to periodic points is one-to-one.

The exercise points to other exercises that I also don't know.

Let $$X$$ be an irreducible shift of finite type and $$f:X\rightarrow Y$$ a sliding block code. Prove that $$f$$ is an embedding if and only if it is one-to-one on the periodic points of $$X$$.

I know the definitions for sofic, finite type, irreducible and sliding block code; but have no idea how to use them. Why things should be different here for sofic shifts and shifts of finite type?

• Glossary: a "sliding block code" is just a continuous equivariant map $f$, i.e., such that $f\circ s_X=s_Y\circ f$ where $s_X$ and $s_Y$ are the shift maps. – YCor Mar 31 at 8:29

This is a fun pair of exercises (the first one you mention is 3.2.9 and the second is 2.3.6a)! For 2.3.6a, recode to a $$1$$-block code $$\phi$$ on an irreducible edge shift $$X$$, suppose that $$x, x' \in X$$ are such that $$x \neq x'$$ but $$\phi(x) = \phi(x')$$. Consider two cases: either $$x_j \neq x_j'$$ for either all $$j > i$$ or all $$j < i$$; or there exist $$j < i < k$$ with $$x_j = x_j'$$, $$x_k = x_k'$$. In either case, try to construct a pair of distinct periodic points whose images agree.
Then, for 3.2.9, here's an idea for a construction: try to construct a sofic shift $$X$$ by labelling an irreducible graph with alphabet $$\{ 0, 1, 2 \}$$, such that the path presenting $$10^m 1$$ can be determined by whether $$m$$ is even or odd, and such that every cycle (other than self-loops) contains a $$2$$. Then code from $$X$$ to $$Y$$ by replacing $$2$$ by $$1$$ and leaving the other symbols the same.