An algorithm for the word entropy

TL;DR

This paper introduces an algorithm to accurately estimate the word entropy of infinite sequences based on their complexity functions, linking combinatorial properties with fractal dimensions.

Contribution

It provides a novel algorithm to compute the supremum of entropy for sequences with complexity bounded by a given exponential growth function.

Findings

Algorithm achieves arbitrary precision in estimating word entropy.

Establishes connections between word entropy and fractal dimensions.

Enables practical computation of entropy bounds for complex sequences.

Abstract

For any infinite word $w$ on a finite alphabet $A$, the complexity function $p_w$ of $w$ is the sequence counting, for each non-negative $n$, the number $p_w(n)$ of words of length $n$ on the alphabet $A$ that are factors of the infinite word $w$ and the the entropy of $w$ is the quantity $E(w)=\lim\limits_{n\to\infty}\frac 1n\log p_w(n)$. For any given function $f$ with exponential growth, Mauduit and Moreira introduced in [MM17] the notion of word entropy $E_W(f) = \sup \{E(w), w \in A^{{\mathbb N}}, p_w \le f \}$ 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 $E_W(f)$ from finitely many values of $f$.