WORDS 2017: Combinatorics on Words pp 157-163

# The Word Entropy and How to Compute It

• Sébastien Ferenczi
• Christian Mauduit
• Carlos Gustavo Moreira
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10432)

## Abstract

The complexity function of an infinite word counts the number of its factors. For any positive function f, its exponential rate of growth $$E_0(f)$$ is $$\lim \limits _{n\rightarrow \infty } \inf \frac{1}{n}\log f(n)$$. We define a new quantity, the word entropy $$E_W(f)$$, as the maximal exponential growth rate of a complexity function smaller than f. This is in general smaller than $$E_0(f)$$, and more difficult to compute; we give an algorithm to estimate it. The quantity $$E_W(f)$$ is used to compute the Hausdorff dimension of the set of real numbers whose expansions in a given base have complexity bounded by f.

## Keywords

Word complexity Positive entropy

## References

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Mauduit, C., Moreira, C.G.: Complexity and fractal dimensions for infinite sequences with positive entropy (2017)Google Scholar

© Springer International Publishing AG 2017

## Authors and Affiliations

• Sébastien Ferenczi
• 1
Email author
• Christian Mauduit
• 1
• Carlos Gustavo Moreira
• 2
1. 1.Aix Marseille Université, CNRS, Centrale Marseille, Institut de Mathématiques de Marseille, I2M - UMR 7373Marseille Cedex 9France
2. 2.Instituto de Matemática Pura e AplicadaRio de JaneiroBrazil