Complexity and fractal dimensions for infinite sequences with positive entropy

TL;DR

This paper investigates the complexity and fractal dimensions of infinite sequences with bounded complexity functions, introducing a new measure called word entropy and linking it to topological entropy.

Contribution

It defines the word entropy for functions bounding complexity, relates it to fractal dimensions, and provides a combinatorial proof connecting it to topological entropy.

Findings

Word entropy equals the topological entropy of the associated subshift.

Fractal dimensions of sequence sets are characterized by word entropy.

Word entropy can be strictly smaller than the exponential growth rate of the complexity function.

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 $f$.