Skip to Main Content

Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Rigidity of critical circle mappings II
HTML articles powered by AMS MathViewer

by Edson de Faria and Welington de Melo
J. Amer. Math. Soc. 13 (2000), 343-370
Published electronically: November 23, 1999


We prove that any two real-analytic critical circle maps with cubic critical point and the same irrational rotation number of bounded type are $C^{1+\alpha }$ conjugate for some $\alpha >0$.
  • Lars V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0357743
  • E. de Faria, Asymptotic rigidity of scaling ratios for critical circle mappings, Ergod. Th. & Dynam. Sys. 19 (1999), 995–1035.
  • E. de Faria and W. de Melo, Rigidity of critical circle mappings I, J. Eur. Math. Soc. 1.
  • Jacek Graczyk and Grzegorz Świa̧tek, Critical circle maps near bifurcation, Comm. Math. Phys. 176 (1996), no. 2, 227–260. MR 1374412
  • 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
  • —, Conjugaison quasi-simétrique des homéomorphismes du cercle a des rotations, Manuscript, 1988.
  • Linda Keen, Dynamics of holomorphic self-maps of $\textbf {C}^*$, Holomorphic functions and moduli, Vol. I (Berkeley, CA, 1986) Math. Sci. Res. Inst. Publ., vol. 10, Springer, New York, 1988, pp. 9–30. MR 955806, DOI 10.1007/978-1-4613-9602-4_{2}
  • M. Lyubich and M. Yampolsky, Dynamics of quadratic polynomials: complex bounds for real maps, Ann. Inst. Fourier (Grenoble) 47 (1997), no. 4, 1219–1255 (English, with English and French summaries). MR 1488251
  • Curtis T. McMullen, Complex dynamics and renormalization, Annals of Mathematics Studies, vol. 135, Princeton University Press, Princeton, NJ, 1994. MR 1312365
  • Curtis T. McMullen, Renormalization and 3-manifolds which fiber over the circle, Annals of Mathematics Studies, vol. 142, Princeton University Press, Princeton, NJ, 1996. MR 1401347, DOI 10.1515/9781400865178
  • Curtis T. McMullen, Self-similarity of Siegel disks and Hausdorff dimension of Julia sets, Acta Math. 180 (1998), no. 2, 247–292. MR 1638776, DOI 10.1007/BF02392901
  • Welington de Melo and Sebastian van Strien, One-dimensional dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 25, Springer-Verlag, Berlin, 1993. MR 1239171, DOI 10.1007/978-3-642-78043-1
  • Dennis Sullivan, Bounds, quadratic differentials, and renormalization conjectures, American Mathematical Society centennial publications, Vol. II (Providence, RI, 1988) Amer. Math. Soc., Providence, RI, 1992, pp. 417–466. MR 1184622
  • Grzegorz Świątek, Rational rotation numbers for maps of the circle, Comm. Math. Phys. 119 (1988), no. 1, 109–128. MR 968483
  • M. Yampolsky, Complex bounds for critical circle maps, Ergod. Th. & Dynam. Sys. 19 (1999), 227–257.
  • J.-C. Yoccoz, Conjugaison analytique des difféomorphismes du cercle, Manuscript, 1989.
Similar Articles
Bibliographic Information
  • Edson de Faria
  • Affiliation: Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, CEP05508-900 São Paulo SP - Brasil
  • MR Author ID: 357550
  • Email:
  • Welington de Melo
  • Affiliation: Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Jardim Botânico, CEP22460-320 Rio de Janeiro RJ - Brasil
  • Email:
  • Received by editor(s): November 9, 1998
  • Received by editor(s) in revised form: September 20, 1999
  • Published electronically: November 23, 1999
  • Additional Notes: This work has been partially supported by the Pronex Project on Dynamical Systems, by FAPESP Grant 95/3187-4 and by CNPq Grant 30.1244/86-3.
  • © Copyright 2000 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 13 (2000), 343-370
  • MSC (2000): Primary 37F25; Secondary 37E10, 30D05, 37F40
  • DOI:
  • MathSciNet review: 1711394