Article PDF
Avoid common mistakes on your manuscript.
References
[Ah1]Ahlfors, L.,Lectures on Quasiconformal Mappings. Van Nostrand, Toronto-New York-London, 1966.
[Ah2] —,Conformal Invariants: Topics in Geometric Function Theory. McGraw-Hill, New York-Düsseldorf-Johannesburg, 1973.
[As]Astala, K., Area distortion of quasiconformal mappings.Acta Math., 173 (1994), 37–60.
[Be]Beardon, A.,Iteration of Rational Functions, Graduate Texts in Math., 132. Springer-Verlag New York, 1991.
[D]Douady, A., Disques de Siegel et anneaux de Herman, inSéminaire Bourbaki, vol. 1988/87, exp. no 677.Astérisque, 152–153 (1987), 151–172.
[DE]Douady, A. &Earle, C. J., Conformally natural extension of homeomorphisms of the circle.Acta Math., 157 (1986), 23–48.
[DH1]Douady, A. &Hubbard, J., On the dynamics of polynomial-like mappings.Ann. Sci. École Norm. Sup., 18 (1985), 287–344.
[DH2] —, A proof of Thurston's topological characterization of rational maps.Acta Math., 171 (1993), 263–297.
[EMc]Earle, C. J. &McMullen, C., Quasiconformal isotopies, inHolomorphic Functions and Moduli, Vol. I (Berkeley, CA, 1986), pp. 143–154. Math. Sci. Res. Inst. Publ., 10. Springer-Verlag, New York-Berlin, 1988.
[F]Faria E. de, Asymptotic rigidity of scaling ratios for critical circle mappings. Preprint.
[FM]Faria, D. de & Melo, W. de, Rigidity of critical circle mappings. In preparation.
[He]Herman, M., Conjugaison quasi-symétrique des difféomorphismes du cercle et applications aux disques singuliers de Siegel. Manuscript, 1986.
[HW]Hardy, G. H. &Wright, E. M.,An Introduction to the Theory of Numbers, 5th edition. Oxford Univ. Press, New York, 1979.
[LV]Lehto, O. &Virtanen, K. I.,Quasiconformal Mappings in the Plane, 2nd edition. Grundlehren Math. Wiss., 126. Springer-Verlag, New York-Heidelberg, 1973.
[Mc1]McMullen, C., Families of rational maps and iterative root-finding algorithms.Ann. of Math., 125 (1987), 467–493.
[Mc2] —, Automorphisms of rational maps, inHolomorphic Functions and Moduli, Vol. I (Berkeley, CA, 1986), pp. 31–60. Math. Sci. Res. Inst. Publ., 10. Springer-Verlag, New York-Berlin, 1988.
[Mc3] —,Complex Dynamics and Renormalization, Ann. of Math. Stud., 135. Princeton Univ. Press, Princeton, NJ, 1994.
[Mc4] —,Renormalization and 3-Manifolds which Fiber over the Circle. Ann. of Math. Stud., 142. Princeton Univ. Press, Princeton, NJ, 1996.
[McS]McMullen, C. & Sullivan, D., Quasiconformal homeomorphisms and dynamics III: The Teichmüller space of a holomorphic dynamical system. To appear inAdv. Math., 1998.
[MN]Manton, N. S. &Nauenberg, M., Universal scaling behavior for iterated maps in the complex plane.Comm. Math. Phys., 89 (1983), 555–570.
[Pe]Petersen, C. L., Local connectivity of some Julia sets containing a circle with an irrational rotation.Acta Math., 177 (1996), 163–224.
[Sh]Shishikura, M., The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets. SUNY Preprint, 1991/7.
[Si]Siegel, C. L., Iteration of analytic functions.Ann. of Math., 43 (1942), 607–612.
[St]Stein, E. M.,Singular Integrals and Differentiability Properties of Functions. Princeton Math. Ser., 30. Princeton Univ. Press, Princeton, NJ, 1970.
[Su]Sullivan, D., Bounds, quadratic differentials, and renormalization conjectures, inAmerican Mathematical Society Centennial Publications, Vol. II (Providence, RI, 1988), pp. 417–466. Amer. Math. Soc., Providence, RI, 1992.
[Sw]Świątek, G., On critical circle homeomorphisms. Preprint, 1997.
[Wi]Widom, M., Renormalization group analysis of quasiperiodicity in analytic maps.Comm. Math. Phys., 92 (1983), 121–136.
Author information
Authors and Affiliations
Additional information
Research partially supported by the NSF.
Rights and permissions
About this article
Cite this article
McMullen, C.T. Self-similarity of Siegel disks and Hausdorff dimension of Julia sets. Acta Math. 180, 247–292 (1998). https://doi.org/10.1007/BF02392901
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02392901