On ordinal ranks of Baire class functions

TL;DR

This paper investigates ordinal ranks on Baire class functions, establishing equivalences between certain ranks and analyzing their multiplicative properties, especially in relation to product functions.

Contribution

It answers key questions about the relationships and properties of ordinal ranks on Baire class functions, extending previous theories.

Findings

Ranks β* and γ* are essentially equivalent for all countable ξ.

Neither β* nor γ* is essentially multiplicative.

Characterization of functions for which β(fg) ≤ ω^ξ when β(g) ≤ ω^ξ.

Abstract

The theory of ordinal ranks on Baire class 1 functions developed by Kechris and Loveau was recently extended by Elekes, Kiss and Vidny\'{a}nszky to Baire class $\xi$ functions for any countable ordinal $\xi\geq1$. 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 $\xi\geq1,$ 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 $\beta(fg)\leq \omega^{\xi}$ whenever $\beta(g)\leq\omega^{\xi}$ for any countable ordinal $\xi.$