Looking at images of the Standard map (see https://people.math.harvard.edu/~knill/technology/chirikov for a Javascript animation) is fascinating because the pictures illustrate lots of unsolved enigmas. The picture shows stability and instability mixed. Invariant curves are obtained by solving the functional equation for smooth functions. KAM theory establishes that. In the movie, I illustrate a bit, how we were trying to look for a simpler proof of this relation. One can see the functional equation also as a fixed point problem for the operator . This is of course not a contraction so that a Banach fixed point theorem does not work. We can also not use an other “easy theorem” like a Lefschetz formula to find fixed points near which is a solution at . The fixed point equation by the way leads to the next video on October 22 about a central limit theorem. When looking at a random variable and averaging two IID copies of it, one gets a new distribution. The Cauchy distribution is a fixed point of this renormalization map. One calls this a stable distribution.

Back to this video: it is organized around the implicit function theorem, one of the most important theorems in mathematics. The video also relates back to the last video about intersection and incidence calculus relations and I will come back to this. It is extremely interesting that when changing the Laplacian to an intersection Laplacian that we do not need any hard implicit function theorem.