Entropy ratio for infinite sequences with positive entropy

TL;DR

This paper investigates the relationship between the complexity growth rate of infinite sequences and their entropy, providing estimates that connect the combinatorial structure of sequences to their exponential growth characteristics.

Contribution

It introduces bounds on word entropy based on the lower exponential growth rate of the complexity function of infinite sequences.

Findings

Estimates of word entropy in terms of exponential growth rates

Characterization of infinite words with bounded complexity functions

Analysis of the combinatorial structure of 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 $w$. For any given function $f$ with exponential growth, we introduced in [MM17] the notion of {\it 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 $f$.