The Herman invariant tori conjecture

TL;DR

This paper proves the Herman invariant tori conjecture by establishing convergence to a new normal form at a critical point of an analytic Hamiltonian under Bruno conditions, confirming the existence of many invariant tori.

Contribution

It introduces a new type of normal form at critical points and simplifies the proof of the Herman invariant tori conjecture using Banach functors.

Findings

Proves convergence to the new normal form under Bruno conditions

Establishes the existence of a positive measure set of invariant tori near the critical point

Provides a simplified, updated proof of the 2012 result

Abstract

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/