On ordinal ranks of Baire class functions

Abstract

The theory of ordinal ranks on Baire class 1 functions developed by Kechris and Loveau was recently extended by Elekes, Kiss and Vidnyánszky to Baire class $\xi$ functions for any countable ordinal $\xi \ge 1$. In this paper, we answer two of the questions raised by them in their paper (Ranks on the Baire class $\xi$ functions, Trans. Amer. Math. Soc. 368(2016), 8111-8143). Specifically, we show that for any countable ordinal $\xi \ge 1,$ the ranks ${\beta }_{\xi }^{\ast }$ and ${\gamma }_{\xi }^{\ast }$ are essentially equivalent, and that neither of them is essentially multiplicative. Since the rank $\beta$ is not essentially multiplicative, we investigate further the behavior of this rank with respect to products. We characterize the functions $f$ so that $\beta (fg)\le {\omega }^{\xi }$ whenever $\beta (g)\le {\omega }^{\xi }$ for any countable ordinal $\xi .$