# Embeddings of subshifts

Consider $(X,\sigma_X)$ and $(Y, \sigma_{Y})$ be subshifts of the one sided shift in two symbols. Assume that $(X,\sigma_X)$ is a transitive subshift of finite type and $(Y, \sigma_{Y})$ is a transitive sofic subshift such that $h_{top}(\sigma_X) < h_{top}(\Sigma_{Y})$.

My question is: What are the conditions (if any) to be sure that there is a proper embedding from X to Y?

It is known that if $(X,\sigma_X)$ and $(Y, \sigma_{Y})$ are both transitive subshifts of finite type then such embedding exists if and only if $h_{top}(\sigma_X) < h_{top}(\sigma_{Y})$ and $Per_n(\sigma_X) \leq Per_n(\sigma_Y)$ for every $n \in \mathbb{N}$ where $Per_n$ denotes the cardinality of the set of points having least period $n$. I believe that $h_{top}(\sigma_X) < h_{top}(\Sigma_{Y})$ is a necessary condition. Also, the examples that I am working with satisfy that $Per(\sigma_X) \cap Per(\sigma_Y) = \emptyset$.

Not sure what your last condition (the intersection of periodic point sets is empty? for some $n$ or for all $n$?) means. You could have $X$ containing the fixed point of all 1s and $Y$ the fixed point of all 0s and the two shifts could even be conjugate. E.g. think of $X$ as the Fibonacci and $Y$ the "flipped Fibonacci", where you interchanged the role of 0s and 1s etc.
• Hi, About the last condition, basically it was a bad attempt (sorry about that) to explain my setup. Actually, in my setup $X$ is not a subset of $Y$ and $X \cap Y = \emptyset$. I'll check the paper, thanks a lot! – Rafael Alcaraz Barrera Mar 31 '15 at 20:55