An algorithm for the word entropy

Abstract

For any infinite word w on a finite alphabet A, the complexity function pw of w is the sequence counting, for each non-negative n, the number pw(n) of words of length n on the alphabet A that are factors of the infinite word w and the entropy of w is the quantity E(w)=limn→∞⁡1nlog⁡pw(n). For any given function f with exponential growth, Mauduit and Moreira introduced in [10] the notion of word entropy EW(f)=sup⁡{E(w),w∈AN,pw≤f} and showed its links with fractal dimensions of sets of infinite sequences with complexity function bounded by f. The goal of this work is to give an algorithm to estimate with arbitrary precision EW(f) from finitely many values of f.