A note on derivatives, expansions and $\Pi^1_1$-ranks

TL;DR

This paper explores variants of $ ext{Pi}^1_1$-ranks, particularly the $ ext{Gamma}$-rank, and their applications to dynamical ranks such as entropy, extending existing techniques in descriptive set theory.

Contribution

It proves a new variant of Kechris's technique for constructing $ ext{Pi}^1_1$-ranks, applicable to the $ ext{Gamma}$-rank, with implications for dynamical systems.

Findings

Proves a variant of Kechris's technique for $ ext{Pi}^1_1$-ranks

Shows applicability to the $ ext{Gamma}$-rank

Connects $ ext{Gamma}$-rank to dynamical ranks like entropy

Abstract

$\Pi_1^1$-ranks are a natural tool for studying coanalytic sets in descriptive set theory. In the book "Classical descriptive set theory", Kechris provided a technique to build $\Pi_1^1$-ranks using derivatives. In this note we will prove a variant of this result that is applicable to the $\Gamma$-rank. Some dynamical ranks, like the entropy rank can be stated in terms of the $\Gamma$-rank.