COST AND DIMENSION OF WORDS OF ZERO TOPOLOGICAL ENTROPY

The (factor) complexity of a language L is defined as a function p(L) (n) which counts for each n the number of words in L of length n. We are interested in whether L is contained in a finite product of the form S-k, where S is a language of strictly lower complexity. In this paper, we focus on languages of zero topological entropy, meaning lim sup(n ->infinity) log p(L)(n)/n = 0. We define the alpha-dimension of a language L as the infimum of integer numbers k such that there exists a language S of complexity O(n(alpha)) such that L subset of S-k. We then define the cost c(L) as the infimum of all real numbers alpha for which the alpha-dimension of L is finite. In particular, the above definitions apply to the language of factors of an infinite word. In the paper, we search for connections between the complexity of a language (or an infinite word) and its dimension and cost, and show that they can be rather complicated.