Cost and dimension of words of zero topological entropy

Abstract

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 p S (n) = O(n α) and a positive integer k with L ⊆ S k. We also define the cost dimension d c (L) as the infimum of the set of all positive integers k such that L ⊆ S k for some language S with p S (n) = O(n c(L)). We are primarily interested in languages L given by the set of factors of an infinite word x = x 0 x 1 x 2 • • • ∈ A N of zero topological entropy, in which case c(L) 2 there exist infinite words x of positive cost and of complexity p x (n) = O(n α).