Semi-strongly irreducible shifts

AbstractIf Z is the group of integers, A a finite alphabet and AZ the set of all functions c:Z→A, the equivalence between pre-injectivity and surjectivity of a local function holds for irreducible shifts of finite type of AZ (see [Pure Math. Appl. 11 (2000) 471–484]). In [Theoret. Comput. Sci. 299 (2003) 477–493] we give a definition of strong irreducibility that, together with the finite type condition, allows us to prove the above equivalence for strongly irreducible shifts of finite type in AΓ, if Γ is an amenable group. In this paper, we define semi-strong irreducibility for a shift. This property allows us to prove the implication “pre-injective ⇒ surjective” for a local function on a semi-strongly irreducible shift of finite type of AΓ, if Γ has nonexponential growth. As a by-product, we prove that the entropy of a proper subshift of a semi-strongly irreducible shift X is strictly smaller than the entropy of X