Rauzy dimension and finite-state dimension

Abstract

In 1976, Rauzy studied two complexity functions, $\underline{\beta}$ and $\overline{\beta}$, for infinite sequences over a finite alphabet. The function $\underline{\beta}$ achieves its maximum precisely for Borel normal sequences, while $\overline{\beta}$ 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, $\underline{\beta}$ and $\overline{\beta}$, and the notions of non-aligned block entropy, $\underline{h}$ and $\overline{h}$, by providing sharp upper and lower bounds for $\underline{h}$ in terms of $\underline{\beta}$, and sharp upper and lower bounds for $\overline{h}$ in terms of $\overline{\beta}$. 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, $\overline{h}$ and $\underline{h}$, in terms of Shannon's conditional entropy. The bounds imply that sequences with $\overline{h} = 0$ coincide with those for which $\overline{\beta} = 0$. We also show that the non-aligned block entropies, $\underline{h}$ and $\overline{h}$, are essentially subadditive.