Cylinder renormalization of siegel disks

Experimental Mathematics

Volumn 16, Issue 2, 2007, Pages 215-226

  Gaidashev, Denis ^a   Yampolsky, Michael ^b  

^a Institutionen for Matematik   (Sweden)

^b UNIVERSITY OF TORONTO   (Canada)

Author keywords

Beltrami equation; Measurable Riemann Mapping Theorem; Renormalization; Siegel disc; Universality

Indexed keywords

EID: 34548327188     PISSN: 10586458     EISSN: 1944950X     Source Type: Journal    
DOI: 10.1080/10586458.2007.10128991     Document Type: Article

References (31)

1. L. Ahlfors. Lectures on Quasiconformal Mappings. Princeton: Van Nostrand-Reinhold, 1966.
2. L. Ahlfors and L. Bers. “Riemann’s Mapping Theorem for Variable Metrics.” Ann. Math. (2) 72: 2 (1960), 385–404.
3. L. Bers. “Quasiconformal Mappings, with Applications to Differential Equations, Function Theory and Topology.” Bull. Amer. Math. Soc. 83 (1977), 1083–1100.
4. 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.
5. 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.
6. A. P. Calderón, A. Zygmund. “On Singular Integrals.” Amer. J. Math. 78 (1956), 289–309.
7. L. Carleson, Th. W. Gamelin. Complex Dynamics. New York: Springer, 1991.
8. P. Daripa. “A Fast Algorithm to Solve Nonhomogeneous Cauchy–Riemann Equations in the Complex Plane.” SIAM J. Sci. Statist. Comput. 13, (1992), 1418-1432.
9. P. Daripa. “A Fast Algorithm to Solve the Beltrami Equation with Applications to Quasiconformal Mappings.” J. Comput. Phys. 106 (1993), 355–365.
10. P. Daripa and D. Mashat. “Singular Integral Transforms and Fast Numerical Algorithms.” Numer. Algor. 18 (1998), 133–157.
11. E. de Faria. “Proof of Universality for Critical Circle Mappings.” PhD diss., CUNY, 1992.
12. E. de Faria. “Asymptotic Rigidity of Scaling Ratios for Critical Circle Mappings. Ergodic Theory Dynam. Systems 19: 4 (1999), 995–1035.
13. A. Fletcher and V. Markovic. Quasiconformal Maps and Teichmüller Theory, Oxford Graduate Texts in Mathematics. New York: Oxford University Press, 2006.
14. D. Gaidashev. “Cylinder Renormalization for Siegel Discs and a Constructive Measurable Riemann Mapping Theorem.” Nonlinearity 20: 3 (2007), 713–741.
15. D. Gaidashev and D. Khmelev. “On Numerical Algorithms for the Solution of a Beltrami Equation.” math.DS/0510516 at Arxiv.org, 2006.
16. 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.
17. 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.
18. M. Lyubich. “Dynamics of Rational Transformations: Topological Picture.” Uspekhi Mat. Nauk 41:4 (1986), 35–95.
19. 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.
20. N. S. Manton and M. Nauenberg. “Universal Scaling Behaviour for Iterated Maps in the Complex Plane.” Commun. Math. Phys. 89 (1983), 555-570.
21. C. McMullen. “Self-Similarity of Siegel Disks and Hausdorff Dimension of Julia Sets.” Acta Math. 180 (1998), 247–292.
22. 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.
23. M. Shishikura. “The Hausdorff Dimension of the Boundary of the Mandelbrot Set and Julia Sets.” Ann. of Math. 43 (1942), 607–612.
24. C. L. Siegel. “Iteration of Analytic Functions.” Ann. Math. 43 (1942), 607–612.
25. A. Stirnemann. “Existence of the Siegel Disc Renormalization Fixed Point.” Nonlinearity 7: 3 (1994), 959–974.
26. 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.
27. M. Widom. “Renormalisation Group Analysis of Quasi-periodicity in Analytic Maps.” Commun. Math. Phys. 92 (1983), 121–136.
28. M. Yampolsky. “Hyperbolicity of Renormalization of Critical Circle maps.” Publ. Math. Inst. Hautes Etudes Sci. 96 (2002), 1–41.
29. M. Yampolsky. “Renormalization Horseshoe for Critical Circle Maps.” Commun. Math. Physics 240, (2003), 75–96.
30. M. Yampolsky. Siegel Disks and Renormalization Fixed Points.” math.DS/0602678 at Arxiv.org, 2006.
31. M. Yampolsky. Programs available online (http://www.math.toronto.edu/yampol/siegel-numerics.tar.bz2), 2007.