  We study a new type of normal form at a critical point of an analytic
Hamiltonian. Under a Bruno condition on the frequency, we prove a convergence
statement to the normal form. Using this result, we prove the Herman invariant
tori conjecture namely the existence of a positive measure set of invariant
tori near the critical point. This paper is an update of the first 2012 proof
of the author. The functional analytic arguments have been simplified using
Banach functors, minor points have been clarified.
  A series of videos is available on the webpage
https://www.agtz.mathematik.uni-mainz.de/category/alg-geom/
