KAM below \mathbf Cⁿ

2021/04/05 by Pöschel, Jürgen
#37J40 #70H08 #70K43 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2104.01866

Abstract

We consider the KAM theory for rotational flows on an n-dimensional torus. We show that if its frequencies are diophantine of type n-1, then Moser's KAM theory with parameters applies to small perturbations of weaker regularity than Cⁿ. Derivatives of order n need not be continuous, but rather L² in a certain strong sense. This disproves the long standing conjecture that Cⁿ is the minimal regularity assumption for KAM to apply in this setting while still allowing for Herman's C^n-ε -counterexamples.

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