Complexity and fractal dimensions for infinite sequences with positive entropy

Carlos Gustavo Moreira, Christian Mauduit

Published 2017-02-24Version 1

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 $w$. One of the goals of this work is to estimate the number of words of length $n$ on the alphabet $A$ that are factors of an infinite word $w$ with a complexity function bounded by a given function $f$ with exponential growth. We describe the combinatorial structure of such sets of infinite words and we introduce a real parameter, the $word\,\, entropy$ $E_W(f)$ associated to a given function $f$. We give estimates on the word entropy $E_W(f)$ in terms of the exponential rate of growth of $f$ and we determine fractal dimensions of sets of infinite sequences with complexity function bounded by $f$ in terms of its word entropy.