Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



$C^1$ 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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • 1. Arnold, V. I. Small denominators I: On mappings of the circle onto itself. Translations AMS(1965), 213-284.
  • 2. Philip L. Boyland, Bifurcations of circle maps: Arnol′d tongues, bistability and rotation intervals, Comm. Math. Phys. 106 (1986), no. 3, 353–381. MR 859816
  • 3. Welington de Melo and Charles Pugh, The 𝐶¹ Brunovský hypothesis, J. Differential Equations 113 (1994), no. 2, 300–337. MR 1297660,
  • 4. Joseph Diestel, Sequences and series in Banach spaces, Graduate Texts in Mathematics, vol. 92, Springer-Verlag, New York, 1984. MR 737004
  • 5. Glen Richard Hall, Resonance zones in two-parameter families of circle homeomorphisms, SIAM J. Math. Anal. 15 (1984), no. 6, 1075–1081. MR 762964,
  • 6. Michael-Robert Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Inst. Hautes Études Sci. Publ. Math. 49 (1979), 5–233 (French). MR 538680
  • 7. Michael-R. Herman, Majoration du nombre de cycles périodiques pour certaines familles de difféomorphismes du cercle, An. Acad. Brasil. Ciênc. 57 (1985), no. 3, 261–263 (French, with English summary). MR 832734
  • 8. Ya. G. Sinaĭ and K. M. Khanin, Smoothness of conjugacies of diffeomorphisms of the circle with rotations, Uspekhi Mat. Nauk 44 (1989), no. 1(265), 57–82, 247 (Russian); English transl., Russian Math. Surveys 44 (1989), no. 1, 69–99. MR 997684,
  • 9. Khinchin, A. Continued Fractions. UMI, Michigan, 1991.
  • 10. Slammert, L. Limit differentiability properties of rational arcs in Arnol'd Tongues, 1998, Aalborg University, Institute of Mathematics, R-98-2019.
  • 11. Jaroslav Stark, Smooth conjugacy and renormalisation for diffeomorphisms of the circle, Nonlinearity 1 (1988), no. 4, 541–575. MR 967471
  • 12. J.-C. Yoccoz, Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation vérifie une condition diophantienne, Ann. Sci. École Norm. Sup. (4) 17 (1984), no. 3, 333–359 (French, with English summary). MR 777374

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 58F03, 58F13, 58F14, 58F11

Retrieve articles in all journals with MSC (2000): 58F03, 58F13, 58F14, 58F11

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