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?