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arXiv · 1803.05533

An algorithm for the word entropy

Carlos Gustavo Moreira·Christian Mauduit·Sébastien Ferenczi
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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 $f$.

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Carlos Gustavo Moreira, Christian Mauduit, Sébastien Ferenczi. 2018-03-14. An algorithm for the word entropy. https://arxiv.org/abs/1803.05533

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