𝐶¹ smoothness of Liouville arcs in Arnol’d tongues

Slammert, Lionel

doi:10.1090/S0002-9939-01-06043-9

$C^1$ smoothness of Liouville arcs in Arnol’d tongues
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Proc. Amer. Math. Soc. 129 (2001), 1817-1823 Request permission

Abstract:

For the generic two parameter family of $C^r$ circle diffeomorphisms of a general form we prove that the bifurcation arcs which correspond to Liouville irrational rotation numbers are $C^1$ smooth. As a consequence, we give an explicit formula for the derivative of all non-resonance arcs. Results of Arnol’d, Herman, and others give greater smoothness for a more restricted class of rotation numbers using KAM techniques.

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