Cost and dimension of words of zero topological entropy

Let A∗ denote the free monoid generated by a finite nonempty set A. In this paper we introduce a new measure of complexity of languages L ⊆ A∗ defined in terms of the semigroup structure on A∗. For each L ⊆ A∗, we define its cost c(L) as the infimum of all real numbers α for which there exist a language S ⊆ A∗ with pS(n) = O(n) and a positive integer k with L ⊆ S.We also define the cost dimension dc(L) as the infimum of the set of all positive integers k such that L ⊆ S for some language S with pS(n) = O(n). We are primarily interested in languages L given by the set of factors of an infinite word x = x0x1x2 · · · ∈ AN of zero topological entropy, in which case c(L) < +∞. We establish the following characterisation of words of linear factor complexity: Let x ∈ AN and L = Fac(x) be the set of factors of x. Then px(n) = Θ(n) if and only c(L) = 0 and dc(L) = 2. In other words, px(n) = O(n) if and only if Fac(x) ⊆ S for some language S ⊆ A of bounded complexity (meaning lim sup pS(n) < +∞). In general the cost of a language L reflects deeply the underlying combinatorial structure induced by the semigroup structure on A∗. For example, in contrast to the above characterisation of languages generated by words of sublinear complexity, there exist non factorial languages L of complexity pL(n) = O(log n) (and hence of cost equal to 0) and of cost dimension +∞. In this paper we investigate the cost and cost dimension of languages defined by infinite words of zero topological entropy. We establish the existence of words of cost zero and finite cost dimension having arbitrarily high polynomial complexity. In contrast we also show that for each α > 2 there exist infinite words x of positive cost and of complexity px(n) = O(n).