Cylinder renormalization for Siegel discs and a constructive measurable Riemann mapping theorem

Nonlinearity

Volumn 20, Issue 3, 2007, Pages 713-741

  Gaidashev, Denis G ^a  

^a Institutionen for Matematik   (Sweden)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33947655858     PISSN: 09517715     EISSN: 13616544     Source Type: Journal    
DOI: 10.1088/0951-7715/20/3/008     Document Type: Article

References (49)

1. Ada Core Technologies 2004 73 Fifth Ave, New York, NY 10003, USA (see also ftp://cs.nyu.edu/pub/gnat)"ftp://cs.nyu.edu/pub/gnat"ftp://cs. nyu.edu/pub/gnat)
2. Ahlfors L 1966 Lectures on Quasiconformal Mappings (Princeton, NJ: Van Nostrand)
3. Ahlfors L and Bers L 1960 Riemann's mapping theorem for variable metrics Ann. Math. 72 385-404
4. Bers L 1977 Quasiconformal mappings, with applications to differential equations, function theory and topology Bull. Am. Math. Soc. 83 1083-100
5. Boyarskii B V 1957 Generalized solutions of systems of differential equations of first order and elliptic type with discontinuous coefficients Mat. Sb. N. S. 43 451-503
6. Boyarskii B V and Iwanier T 1974 Quasiconformal mappings and nonlinear elliptic equations in two variables: I Bull. Acad. Polon. Sci., Sér. Math. Astron. Phys. 12 473-78
7. Boyarskii B V and Iwanier T 1974 Quasiconformal mappings and nonlinear elliptic equations in two variables: II Bull. Acad. Polon. Sci., Sér. Math. Astron. Phys. 12 479-84
8. Calderon A P and Zygmund A 1956 On singular integrals Am. J. Math. 78 289-309
9. Carleson L and Gamelin Th W 1991 Complex Dynamics (Berlin: Springer)
10. Daripa P 1992 A fast algorithm to solve non-homogeneous Cauchy-Riemann equations in the complex plane SIAM J. Sci. Stat. Comput. 13 1418-32
11. Daripa P 1993 A fast algorithm to solve the Beltrami equation with applications to quasiconformal mappings J. Comput. Phys. 106 355-65
12. Daripa P and Mashat D 1998 Singular integral transforms and fast numerical algorithms Numer. Algorithms 18 133-57
13. de Faria E 1992 Proof of universality for critical circle mappings Thesis CUNY, New York
14. de Faria E 1999 Asymptotic rigidity of scaling ratios for critical circle mappings Ergod. Theory Dyn. Systems 19 995-1035
15. de Faria E and de Melo W 1999 Rigidity of critical circle mappings: I J. Eur. Math. Soc. (JEMS) 1 339-92
16. de Faria E and de Melo W 2000 Rigidity of critical circle mappings: II J. Am. Math. Soc. 13 334-70
17. de la Llave R and Rana D 1990 Accurate strategies for small denominator problems Bull. Am. Math. Soc. 22 85-90
18. Douady A 1987 Disques de Siegel et anneaux de Herman Asterisque 152-153 151-72 (Seminar Bourbaki)
19. Eckmann J-P, Koch H and Wittwer P 1982 Existence of a fixed point of the doubling transformation for area-preserving maps of the plane Phys. Rev. A 26 720-22
20. Eckmann J-P, Koch H and Wittwer P 1984 A computer-assisted proof of universality for area-preserving maps Mem. Am. Math. Soc. 47 1-121
21. Gaidashev D and Khmelev D 2005 On numerical algorithms for the solution of the Beltrami equation Preprint math.NA/0510516
22. Gaidashev D and Yampolsky M 2006 Cylinder renormalization of Siegel disks Exp. Math. at press (Preprint math.DS/0603008)
23. Grafakos L 1997 Best bounds for the Hilbert transform on Math. Res. Lett. 4
24. IEEE 1985 The IEEE Standard for Binary Floating-Point Arithmetic ANSI/IEEE Std 754-1985
25. Koch H 2002 On the renormalization of Hamiltonian flows, and critical invariant tori Discrete Contin. Dyn. Syst. 8 633-46
26. Koch H 2004a A renormalization group fixed point associated with the breakup of golden invariant tori Discrete Contin. Dyn. Syst. 11 881-909
27. Koch H 2004b Existence of critical invariant tori Preprint mp_arc 04-210
28. Koch H, Schenkel A and Wittwer P 1996 Computer-assisted proofs in analysis and programming in logic: a case study SIAM Rev. 38 565-604
29. Lanford O E 1982 Computer-assisted proof of the Feigenbaum conjectures Bull. Am. Meteorol. Soc. (New Series) 6 427-34
30. Lanford O E 1984 Computer-assisted proofs in analysis Physica A 124 465-70
31. Lebedev N N 1972 Special Functions and Their Applications (New York: Dover)
32. Lyubich M 1995 Renormalization ideas in conformal dynamics Cambridge Seminar "Current Developments in Math.", May 1995 (Cambridge, MA: International Press) pp 155-84
33. Lyubich M 1999 Feigenbaum-Coullet-Tresser universality and Milnor's hairiness conjecture Ann. Math. 149 319-420
34. Manton N S and Nauenberg M 1983 Universal scaling behaviour for iterated maps in the complex plane Commun. Math. Phys. 89 555-70
35. Markovic V 2003 Quasiconformal maps Lecture Notes available at http://www.maths.warwick.ac.uk/∼fletcher/qcmaps.pdf
36. McMullen C 1998 Self-similarity of Siegel disk and Hausdorff dimension of Julia sets Acta Math. 180 247-92
37. McMullen C 2000 Complex dynamics and renormalization Ann. Math. Studies vol 135 (Princeton, NJ: Princeton University Press)
38. Mestel B D 1989 A computer-assisted proof of universality for cubic critical maps of the circle with golden mean rotation number PhD Thesis Princeton University
39. Siegel C L 1942 Iterations of analytic functions Ann. Math. 43 607-12
40. Shishikura M 1998 The Hausdorff dimension of the boundary of the Mandelrot set and Julia sets Ann. Math. 147 225-67
41. Stirnemann A 1994 Existence of the Siegel disk renormalization fixed point Nonlinearity 7 959-74
42. Sullivan D 1986 Quasiconformal homeomorphism and dynamics, topology and geometry Proc. ICM-86 (Berkeley) vol II pp 1216-28
43. Sullivan D 1992 Bounds, quadratic differentials and renormalization conjectures AMS Centennial Publications 2: Mathematics into Twenty-First Century
44. Taft S T J and Duff R A (ed) 1995 Ada 95 Reference Manual: Language and Standard Libraries, International Standard ISO/IEC 8652:1995(E) (Lecture Notes in Computer Science) vol 1246 (Berlin: Springer)
45. Widom M 1983 Renormalization group analysis of quasi-periodicity in analytic maps Commun. Math. Phys. 92 121-36
46. Yampolsky M 2001 The attractor of renormalization and rigidity of towers of critical circle maps Commun. Math. Phys. 218 537-68
47. Yampolsky M 2002 Hyperbolicity of renormalization of critical circle maps Publ. Math. Inst. Hautes Etudes Sci. 96 1-41
48. Yampolsky M 2003 Global renormalization horseshoe for critical circle maps Commun. Math. Phys. 240 75-96
49. Yampolsky M 2005 Siegel disks and renormalization fixed points Preprint math.DS/0602678