On the entropy of Z subshifts of finite type

Let n be a positive integer and denote < n >= {1, ..., n}. We view < n > as an alphabet on n letters. Denote by < n > d the set of all mappings of Z to the set < n >. By extending the Hamming metric on < n > × < n > to < n >Zd one obtains that < n > d is a compact metric space. The group Z acts as a group of (translation) automorphisms on < n > d . A set S ⊂ Z which is closed and invariant under the action of Z is called a subshift. S is called a subshift of finite type (SFT) if there is a finite set of finite admissible configurations which generates S under the action of Z. More presicely, let F ⊂ Z be a finite set. Assume that P ⊂< n > . Then (F, P ) defines a folowing Z-SFT S. For each a ∈ Z let F + a ⊂ Z be the corresponding translation of F . Then x ∈ S iff for each a ∈ Z, πF+a(x), the projection of x on the set F + a, is in P . See for example [Sch, Ch. 5]. The case of Z action, i.e. d = 1, is well understood. In that case, it is relatively easily to decide whether S is empty or not. Moreover, the topological entropy h(S) of the restriction of the standard shift to S is a logarithm of an algebraic integer ρ(F, P ). The number ρ(P, F ) is a the spectral radius of certain 0− 1 square matrix induced by (F, P ). Furthermore, the topological entropy h(S) is equal to the rate of growth of the number of periodic points. The case d > 1 is much more complicated. First, the problem wether S is an empty set or not is undecidable. This result for d = 2 goes back to Berger [Ber]. See also [K-M-W] and [Rob]. Second, there exists a SFT S 6= ∅ which does not have periodic points. Moreover, in the case where S 6= ∅ the topological entropy may be uncomputable, see [H-K-C] and [Gab]. The object of this paper to show that contrary to these results one has a natural and a simple criterion which either determines that S = ∅ or calculates the topological entropy of S 6= ∅. There is no contradition to the uncomputability of h(S) because we can not estimate the rate of convergence of our sequence. However, if we introduce a symmetry in Z we can estimate the rate of convergence of our sequence. Moreover, in this case h(S) is the rate of growth the number of periodic points. Our main tool is to view a Z-SFT as a matrix SFT. See [M-P1,M-P2]. In fact our methods are very close to the methods of [M-P1,M-P2]. We now describe briefly the content of the paper. In §1 we define combinatorial entropy of Z-SFT. It can is computed by finite configurations. We then observe, using Köning’s method, that Z-SFT is nonempty iff every finite configuration is nonempty. In §2 we show that the combinatorial entropy is equal to the topological entropy of Z-SFT.