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

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.

arXiv admin note: text overlap with arXiv:2107.09263