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$C^1$ smoothness of Liouville arcs in Arnol'd tongues

Author: Lionel Slammert
Journal: Proc. Amer. Math. Soc. 129 (2001), 1817-1823
MSC (2000): Primary 58F03, 58F13, 58F14, 58F11
Published electronically: January 23, 2001
MathSciNet review: 1814115
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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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Additional Information

Lionel Slammert
Affiliation: Department of Mathematics and Applied Mathematics, University of the Western Cape, Bellville, 7535, South Africa
Address at time of publication: Faculty of Applied Science, Cape Technikon, Cape Town 2000, South Africa

Received by editor(s): August 31, 1999
Published electronically: January 23, 2001
Additional Notes: The author thanks the Department of Mathematics at Boston University for a research fellowship that enabled him to do this research.
Communicated by: Michael Handel
Article copyright: © Copyright 2001 American Mathematical Society