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The Word Entropy and How to Compute It

  • Sébastien FerencziEmail author
  • 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.: Generalized Hausdorff dimensions of sets of real numbers with zero entropy expansion. Ergodic Theor. Dynam. Syst. 32(3), 1073–1089 (2012). http://dx.doi.org/10.1017/S0143385711000137
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    Mauduit, C., Moreira, C.G.: Complexity and fractal dimensions for infinite sequences with positive entropy (2017)Google Scholar

Copyright information

© 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

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