On ordinal ranks of Baire class functions

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 $xigeq1$. 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 $xigeq1,$ 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 $beta(g)leqomega^{xi}$ for any countable ordinal $xi.$