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Original Article

Cylinder Renormalization of Siegel Disks

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Pages 215-226 | Published online: 30 Jan 2011

  • L. Ahlfors. Lectures on Quasiconformal Mappings. Princeton: Van Nostrand-Reinhold, 1966.
  • L. Ahlfors and L. Bers. "Riemann's Mapping Theorem for Variable Metrics." Ann. Math. (2) 72:2 (1960), 385–404.
  • L. Bers. "Quasiconformal Mappings, with Applications to Differential Equations, Function Theory and Topology." Bull. Amer. Math. Soc. 83 (1977), 1083–1100.
  • B. V. Boyarskii. "Generalized Solutions of Systems of Differential Equations of First Order and Elliptic Type with Discontinuous Coefficients." Mat. Sb. N. S. 43:4 (1957), 451–503.
  • B. V. Boyarskii and T. Iwanier. "Quasiconformal Mappings and Nonlinear Elliptic Equations in Two Variables, I and II." Bull. Acad. Polon. Sci., Sér. Math. Astronom. Phys. 12 (1974), 473–478 and 479–484.
  • A. P. Calderón, A. Zygmund. "On Singular Integrals." Amer. J. Math. 78 (1956), 289–309.
  • L. Carleson, Th. W. Gamelin. Complex Dynamics. New York: Springer, 1991.
  • P. Daripa. "A Fast Algorithm to Solve Nonhomogeneous Cauchy–Riemann Equations in the Complex Plane." SIAM J. Sci. Statist. Comput. 13, (1992), 1418–1432.
  • P. Daripa. "A Fast Algorithm to Solve the Beltrami Equation with Applications to Quasiconformal Mappings." J. Comput. Phys. 106 (1993), 355–365.
  • P. Daripa and D. Mashat. "Singular Integral Transforms and Fast Numerical Algorithms." Numer. Algor. 18 (1998), 133–157.
  • E. de Faria. "Proof of Universality for Critical Circle Mappings." PhD diss., CUNY, 1992.
  • E. de Faria. "Asymptotic Rigidity of Scaling Ratios for Critical Circle Mappings. Ergodic Theory Dynam. Systems 19:4 (1999), 995–1035.
  • A. Fletcher and V. Markovic. Quasiconformal Maps and Teichmüller Theory, Oxford Graduate Texts in Mathematics. New York: Oxford University Press, 2006.
  • D. Gaidashev. "Cylinder Renormalization for Siegel Discs and a Constructive Measurable Riemann Mapping Theorem." Nonlinearity 20:3 (2007), 713–741.
  • D. Gaidashev and D. Khmelev. "On Numerical Algorithms for the Solution of a Beltrami Equation." math.DS/0510516 at Arxiv.org, 2006.
  • O. E. Lanford. "Renormalization Group Methods for Critical Circle Mappings with General Rotation Number." In VIIIth International Congress on Mathematical Physics (Marseille, 1986), 532–536. Singapore: World Scientific, 1987.
  • O. E. Lanford. "Renormalization Group Methods for Critical Circle Mappings. Nonlinear Evolution and Chaotic Phenomena." NATO Adv. Sci. Inst. Ser. B: Phys. 176 (1988), 25–36.
  • M. Lyubich. "Dynamics of Rational Transformations: Topological Picture." Uspekhi Mat. Nauk 41:4 (1986), 35–95.
  • R. S. MacKay and I. C. Persival. "Universal Small-Scale Structure near the Boundary of Siegel Disks of Arbitrary Rotation Number." Physica 26D (1987), 193–202.
  • N. S. Manton and M. Nauenberg. "Universal Scaling Behaviour for Iterated Maps in the Complex Plane." Commun. Math. Phys. 89 (1983), 555–570.
  • C. McMullen. "Self-Similarity of Siegel Disks and Hausdorff Dimension of Julia Sets." Acta Math. 180 (1998), 247–292.
  • W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling. Numerical Recipes in Fortran. The Art of Scientific Computing. Cambridge: Cambridge University Press, 1992.
  • M. Shishikura. "The Hausdorff Dimension of the Boundary of the Mandelbrot Set and Julia Sets." Ann. of Math. 43 (1942), 607–612.
  • C. L. Siegel. "Iteration of Analytic Functions." Ann. Math. 43 (1942), 607–612.
  • A. Stirnemann. "Existence of the Siegel Disc Renormalization Fixed Point." Nonlinearity 7:3 (1994), 959–974.
  • S. T. J. Taft and R. A. Duff (editors). Ada 95 Reference Manual: Language and Standard Libraries, International Standard ISO/IEC 8652:1995(E). Lecture Notes in Computer Science, 1246. Berlin: Springer-Verlag, 1995.
  • M. Widom. "Renormalisation Group Analysis of Quasi-periodicity in Analytic Maps." Commun. Math. Phys. 92 (1983), 121–136.
  • M. Yampolsky. "Hyperbolicity of Renormalization of Critical Circle maps." Publ. Math. Inst. Hautes Etudes Sci. 96 (2002), 1–41.
  • M. Yampolsky. "Renormalization Horseshoe for Critical Circle Maps." Commun. Math. Physics 240, (2003), 75–96.
  • M. Yampolsky. Siegel Disks and Renormalization Fixed Points." math.DS/0602678 at Arxiv.org, 2006.
  • M. Yampolsky. Programs available online (http://www.math.toronto.edu/yampol/siegel-numerics. tar.bz2), 2007. http://www.math.toronto.edu/yampol/siegel-numerics.tar.bz2

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