Recently, several authors have studied maps where a function, describing the
local diffusion matrix of a diffusion process with a linear drift towards an
attraction point, is mapped into the average of that function with respect to
the unique invariant measure of the diffusion process, as a function of the
attraction point. Such mappings arise in the analysis of infinite systems of
diffusions indexed by the hierarchical group, with a linear attractive
interaction between the components. In this context, the mappings are called
renormalization transformations. We consider such maps for catalytic
Wright-Fisher diffusions. These are diffusions on the unit square where the
first component (the catalyst) performs an autonomous Wright-Fisher diffusion,
while the second component (the reactant) performs a Wright-Fisher diffusion
with a rate depending on the first component through a catalyzing function. We
determine the limit of rescaled iterates of renormalization transformations
acting on the diffusion matrices of such catalytic Wright-Fisher diffusions.Comment: 65 pages, 3 figure