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
HTML articles powered by AMS MathViewer

by Edson de Faria and Welington de Melo

J. Amer. Math. Soc. 13 (2000), 343-370

DOI: https://doi.org/10.1090/S0894-0347-99-00324-0

Published electronically: November 23, 1999

PDF | Request permission

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 $\alpha >0$.

References

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.