Entropy ratio for infinite sequences with positive entropy

Abstract

The complexity function of an infinite word $w$ on a finite alphabet $A$ is the sequence counting, for each non-negative $n$, the number of words of length $n$ on the alphabet $A$ that are factors of the infinite word $w$. For any given function $f$ with exponential growth, we introduced in [MM17] the notion of {\it word entropy} $E_W(f)$ associated to $f$ and we described the combinatorial structure of sets of infinite words with a complexity function bounded by $f$. The goal of this work is to give estimates on the word entropy $E_W(f)$ in terms of the limiting lower exponential growth rate of $f$.

14 pages; the original paper "Complexity and fractal dimensions for infinite sequences with positive entropy", which is the first version of arXiv:1702.07698 was divided in two smaller papers - the first one, with the same title, is now the second version of arXiv:1702.07698, and the second paper is the present one