On ordinal ranks of Baire class functions
Volume 247 / 2019
Fundamenta Mathematicae 247 (2019), 109-130
MSC: Primary 26A21; Secondary 03E15, 54H05.
DOI: 10.4064/fm616-3-2019
Published online: 5 September 2019
Abstract
The theory of ordinal ranks on Baire class 1 functions developed by Kechris and Loveau was extended by Elekes, Kiss and Vidnyánszky (2016) to Baire class $\xi $ functions for any countable ordinal $\xi \geq 1$. We answer two of the questions raised by them. Specifically, we show that for any countable ordinal $\xi \geq 1,$ 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 its behavior with respect to products. We characterize the functions $f$ such that $\beta (fg)\leq \omega ^{\xi }$ whenever $\beta (g)\leq \omega ^{\xi }$ for any countable ordinal $\xi .$
