An abstract KAM theorem (Q2801860)

The article provides an abstract (categorical) setting for iterative processes which are used to prove normal form theorems or to solve functional equations. In general, to obtain such normal forms, one needs to prove the convergence of iterations coming from compositions of changes of local coordinates. The main theorem is formulated in terms of bounded morphisms of Arnold spaces. One important application is the proof of stability, under perturbations, of invariant tori for some Hamiltonian system (KAM theorem). Another remarkable consequence pointed out by the author is his proof of the Herman conjecture about the measure of the set of invariant tori.