Complexity and fractal dimensions for infinite sequences with positive entropy

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 $w$. The goal 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 and to describe the combinatorial structure of such sets of infinite words. We introduce a real parameter, the {\it word entropy} $E_W(f)$ associated to a given function $f$ and we determine the fractal dimensions of sets of infinite sequences with complexity function bounded by $f$ in terms of its word entropy. We present a combinatorial proof of the fact that $E_W(f)$ is equal to the topological entropy of the subshift of infinite words whose complexity is bounded by $f$ and we give several examples showing that even under strong conditions on $f$, the word entropy $E_W(f)$ can be strictly smaller than the limiting lower exponential growth rate of $f$.