Rauzy complexity and block entropy

Abstract

In 1976, Rauzy studied two complexity functions, $\underset{\_}{\beta }$ and $\bar{\beta }$, for infinite sequences over a finite alphabet. The function $\underset{\_}{\beta }$ achieves its maximum precisely for Borel normal sequences, while $\bar{\beta }$ reaches its minimum for sequences that, when added to any Borel normal sequence, result in another Borel normal sequence. We establish a connection between Rauzy's complexity functions, $\underset{\_}{\beta }$ and $\bar{\beta }$, and the notions of non-aligned block entropy, $\underset{\_}{h}$ and $\bar{h}$, by providing sharp upper and lower bounds for $\underset{\_}{h}$ in terms of $\underset{\_}{\beta }$, and sharp upper and lower bounds for $\bar{h}$ in terms of $\bar{\beta }$. We adopt a probabilistic approach by considering an infinite sequence of random variables over a finite alphabet. The proof relies on a new characterization of non-aligned block entropies, $\bar{h}$ and $\underset{\_}{h}$, in terms of Shannon's conditional entropy. The bounds imply that sequences with $\bar{h}=0$ coincide with those for which $\bar{\beta }=0$. We also show that the non-aligned block entropies, $\underset{\_}{h}$ and $\bar{h}$, are essentially subadditive.