Ranks on the Baire class ξ functions

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 complex-ity. We extend this theory to the case of Baire class ξ functions, and generalize most of the results from the Baire class 1 case. As an applcation, we solve a problem concerning the so called solvability cardinals of systems of difference equations, arising from the theory of geometric equidecomposability. We also show that certain other very natural generalizations of the ranks of Kechris and Louveau surprisingly turn out to be bounded in ω1