Sliding block code on irreducible sofic shift

Question

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?

Answer 1

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.

EDIT: I've provided an answer because I didn't notice I was on MO rather than Math.SE, but this question should probably be on Math.SE rather than MO. It's an exercise in an introductory textbook, albeit a textbook with some rather tricky exercises.