Subshifts of $(A^Z , \sigma )$

Let A be a finite set which we will call an alphabet. We assign the discrete topology to A. A^Z is the space obtained with the product topology and is called the full shift. This topology is generated by cylinder sets: $C_m [a_0,\dots ,a_k] = \{x\in A^Z\,\vert\, x_m=a_0, \dots ,x_{m+k}=a_k \}$. So, a point in the full shift is a bi-infinite sequence of elements in A. The shift map is a map $\sigma :A^Z\rightarrowA^Z$ defined with $(\sigma (x))_i = x_{i+1},\,\, i\in Z$. The shift map is continuous and in fact a homeomorphism on A^Z. $(S,\sigma \vert S)$ is called a subshift or symbolic system of A^Z if S is a closed $\sigma$-invariant subset of A^Z. This means that there is a set F(S) of finite words in the alphabet A such that: $x\in S$ iff $\forall k,m \ge 0 \,\,x_k x_{k+1} \dots x_ {k+m} \in F(S)$.

Definition The factor set or factor language of a sofic system S is the set

$$F(S)=\{\,a_0\dots a_m \,\vert\,a_i\in A,\,\exists x\in S,\exists k,\,x_k\dots x_{k+m}=a_0\dots a_m\,\}$$

We will call the elements of F(S) factors and we will use just S to refer either to S or to $(S,\sigma \vert S)$.

Examples Let G=(V,E) be a finite graph with V as a set of vertices and E as a set of edges. Let A=V, and $S\subset A^Z$ be such that $x\in S$ iff $\forall i\in Z$ there is an edge in G from x_i to x_i+1. Then S is a subshift of A^Z. We can also consider A=E and $S\subset A^Z$ is such that $x\in S$ iff $\forall i\in Z$, x_i x_i+1 is a path in G, i.e. the target of x_i is the source of x_i+1. In this case the factor language of S is the set of all finite paths in G.