This is a set of rough, incomplete notes formatted in July/August of 2021 that I have intended to recheck, refine, reorganize, and extend for some time now but simply haven’t found the time to do so although some parts are superseded by some of my posts dated since then. Topics addressed are Lie infinitesimal generators, related flow equations and differential equations, an approach by Schröder to finding zeros of functions and functional compositional iteration, the Maupertuis action and variational principles, and renormalization group flow. Maybe someday, I’ll find the time to both refine and extend these notes.
In the Flow: Infinigins, fixed point equations, and flows in functional iteration and renormalization
(pdf, large file)
Related:
My answer to the MO-Q “Do complex iterates of functions have any meaning?”
Shadows of Simplicity 