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Entropy ratio for infinite sequences with positive entropy

Published online by Cambridge University Press:  10 August 2018

CHRISTIAN MAUDUIT  and
CARLOS GUSTAVO MOREIRA
CHRISTIAN MAUDUIT
Affiliation:
Université d’Aix-Marseille et Institut Universitaire de France, Institut de Mathématiques de Marseille, UMR 7373 CNRS, 163, avenue de Luminy, 13288 Marseille Cedex 9, France
CARLOS GUSTAVO MOREIRA
Affiliation:
Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, 22460-320 Rio de Janeiro, RJ, Brazil
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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 [Complexity and fractal dimensions for infinite sequences with positive entropy. Commun. Contemp. Math. to appear] the notion of 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 $f$.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 3 , March 2020 , pp. 751 - 762
Copyright
© Cambridge University Press, 2018 

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