Complexity and fractal dimensions for infinite sequences with positive entropy

Abstract

The complexity function of an infinite word [Formula: see text] on a finite alphabet [Formula: see text] is the sequence counting, for each non-negative [Formula: see text], the number of words of length [Formula: see text] on the alphabet [Formula: see text] that are factors of the infinite word [Formula: see text]. The goal of this work is to estimate the number of words of length [Formula: see text] on the alphabet [Formula: see text] that are factors of an infinite word [Formula: see text] with a complexity function bounded by a given function [Formula: see text] with exponential growth and to describe the combinatorial structure of such sets of infinite words. We introduce a real parameter, the word entropy [Formula: see text] associated to a given function [Formula: see text] and we determine the fractal dimensions of sets of infinite sequences with complexity function bounded by [Formula: see text] in terms of its word entropy. We present a combinatorial proof of the fact that [Formula: see text] is equal to the topological entropy of the subshift of infinite words whose complexity is bounded by [Formula: see text] and we give several examples showing that even under strong conditions on [Formula: see text], the word entropy [Formula: see text] can be strictly smaller than the limiting lower exponential growth rate of [Formula: see text].