Perturbative analysis of the breathing circle billiard map combined with a quantitative Mather converse-KAM criterion excludes invariant Lipschitz graphs and establishes positive topological entropy for sufficiently small angular momentum.
MacKay, J
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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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Breaking of invariant curves: from the Fermi-Ulam map to the breathing circle billiard
Perturbative analysis of the breathing circle billiard map combined with a quantitative Mather converse-KAM criterion excludes invariant Lipschitz graphs and establishes positive topological entropy for sufficiently small angular momentum.
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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.