$C^1$ smoothness of Liouville arcs in Arnolâd tongues
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- by Lionel Slammert PDF
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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 Arnolâd, Herman, and others give greater smoothness for a more restricted class of rotation numbers using KAM techniques.References
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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
- Email: lslammert@ctech.ac.za
- 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
- © Copyright 2001 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 129 (2001), 1817-1823
- MSC (2000): Primary 58F03, 58F13, 58F14, 58F11
- DOI: https://doi.org/10.1090/S0002-9939-01-06043-9
- MathSciNet review: 1814115