Ranks on the Baire class $\xi$ functions

TL;DR

This paper extends the theory of natural ranks on Baire class 1 functions to Baire class  functions, generalizing previous results, removing compactness assumptions, and applying to difference equations and bounded functions.

Contribution

It generalizes the ranks to Baire class  functions, removes compactness assumptions, and applies the theory to solve problems in difference equations and bounded functions.

Findings

Most results from Baire class 1 extend to class .

Certain natural ranks are bounded in .

All ranks satisfying natural properties coincide for bounded functions.

Abstract

In 1990 Kechris and Louveau developed the theory of three very natural ranks on the Baire class $1$ functions. A rank is a function assigning countable ordinals to certain objects, typically measuring their complexity. We extend this theory to the case of Baire class $\xi$ functions, and generalize most of the results from the Baire class 1 case. We also show that their assumption of the compactness of the underlying space can be eliminated. As an application, we solve a problem concerning the so called solvability cardinals of systems of difference equations, arising from the theory of geometric decompositions. We also show that certain other very natural generalizations of the ranks of Kechris and Louveau surprisingly turn out to be bounded in $\omega_1$. Finally, we prove a general result showing that all ranks satisfying some natural properties coincide for bounded functions.