A weighted Sobolev space on the torus has sharp regularity that lets KAM theory apply to perturbations with classical differentiability only up to floor(n/2) while higher weak derivatives remain unbounded.
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Fixed point theorems extending Zehnder's analytic Nash-Moser theorem show Bruno conditions ensure a positive measure set of invariant tori.
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Sharp regularity of a weighted Sobolev space over $ \mathbb{T}^n $ and its relation to finitely differentiable KAM theory
A weighted Sobolev space on the torus has sharp regularity that lets KAM theory apply to perturbations with classical differentiability only up to floor(n/2) while higher weak derivatives remain unbounded.
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Fixed point theorems for small divisors problems
Fixed point theorems extending Zehnder's analytic Nash-Moser theorem show Bruno conditions ensure a positive measure set of invariant tori.