Rigidity of critical circle mappings II

de Faria, Edson; de Melo, Welington

doi:10.1090/S0894-0347-99-00324-0

Rigidity of critical circle mappings II

Authors: Edson de Faria and Welington de Melo
Journal: J. Amer. Math. Soc. 13 (2000), 343-370
MSC (2000): Primary 37F25; Secondary 37E10, 30D05, 37F40
DOI: https://doi.org/10.1090/S0894-0347-99-00324-0
Published electronically: November 23, 1999
MathSciNet review: 1711394
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: 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 $\alpha >0$ .

[1] L. Ahlfors, Conformal invariants, McGraw-Hill, 1973. MR 50:10211
[2] E. de Faria, Asymptotic rigidity of scaling ratios for critical circle mappings, Ergod. Th. & Dynam. Sys. 19 (1999), 995-1035.
[3] E. de Faria and W. de Melo, Rigidity of critical circle mappings I, J. Eur. Math. Soc. 1.
[4] J. Graczyk and G. Swiatek, Critical circle maps near bifurcation, Commun. Math. Phys. 176 (1996), 227-260. MR 96j:58053
[5] M. Herman, Sur la conjugaison differentiable des difféomorphismes du cercle a des rotations, Publ. Math. IHES 49 (1979), 5-234. MR 81h:58039
[6] -, Conjugaison quasi-simétrique des homéomorphismes du cercle a des rotations, Manuscript, 1988.
[7] L. Keen, Dynamics of holomorphic self-maps of ${\mathbb{C}}^{*}$ , Holomorphic Functions and Moduli I, Eds. D. Drasin et al., Springer-Verlag, New York, 1988. MR 90e:58075
[8] M. Lyubich and M. Yampolsky, Dynamics of quadratic polynomials: complex bounds for real maps, Ann. Inst. Fourier (Grenoble) 47 (1997), 1219-1255. MR 98m:58113
[9] C. McMullen, Complex dynamics and renormalization, Annals of Math. Studies 135 (1994). MR 96b:58097
[10] -, Renormalization and 3-manifolds which fiber over the circle, Annals of Math. Studies 142 (1996). MR 97f:57022
[11] -, Self-similarity of Siegel disks and Hausdorff dimension of Julia sets, Acta Math. 180 (1998), 247-292. MR 99f:58172
[12] W. de Melo and S. van Strien, One-dimensional Dynamics, Springer-Verlag, Berlin and New York, 1993. MR 95a:58035
[13] D. Sullivan, Bounds, quadratic differentials and renormalization conjectures, Mathematics into the Twenty-First Century, Amer. Math. Soc. Centennial Publication, vol.2, Amer. Math. Soc., Providence, RI, 1991. MR 93k:58194
[14] G. Swiatek, Rational rotation numbers for maps of the circle, Commun. Math. Phys. 119 (1988), 109-128. MR 90h:58077
[15] M. Yampolsky, Complex bounds for critical circle maps, Ergod. Th. & Dynam. Sys. 19 (1999), 227-257. CMP 99:09
[16] J.-C. Yoccoz, Conjugaison analytique des difféomorphismes du cercle, Manuscript, 1989.

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 37F25, 37E10, 30D05, 37F40

Retrieve articles in all journals with MSC (2000): 37F25, 37E10, 30D05, 37F40

Additional Information

Edson de Faria
Affiliation: Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, CEP05508-900 São Paulo SP - Brasil
Email: edson@ime.usp.br

Welington de Melo
Affiliation: Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Jardim Botânico, CEP22460-320 Rio de Janeiro RJ - Brasil
Email: demelo@impa.br

DOI: https://doi.org/10.1090/S0894-0347-99-00324-0
Keywords: Holomorphic pairs, complex bounds, uniform twist, rigidity
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.
Article copyright: © Copyright 2000 American Mathematical Society