Dynamics of the universal area-preserving map associated with period-doubling: Stable sets

Journal of Modern Dynamics

Volumn 3, Issue 4, 2009, Pages 555-587

  Gaidashev, Denis ^a   Johnson, Tomas ^a  

^a UPPSALA UNIVERSITY   (Sweden)

Author keywords

Area preserving maps; Period doubling; Renormalization; Rigidity; Rigorous computations

Indexed keywords

EID: 76349125596     PISSN: 19305311     EISSN: None     Source Type: Journal    
DOI: 10.3934/jmd.2009.3.555     Document Type: Article

References (53)

1. J. J. Abad and H. Koch, Renormalization and periodic orbits for Hamiltonian flows, Comm. Math. Phys., 212 (2000), 371-394.
2. J. J. Abad, H. Koch and P. Wittwer, A renormalization group for Hamiltonians: Numerical results, Nonlinearity, 11 (1998), 1185-1194.
3. G. Benettin et al, Universal properties in conservative dynamical systems, Lettere al Nuovo Cimento, 28 (1980), 1-4.
4. T. Bountis, Period doubling bifurcations and universality in conservative systems, Physica D, 3 (1981),577-589.
5. CAPD-Computer Assisted Proofs in Dynamics Group, a C++ package for rigorous numerics, http://capd.wsb-nlu.edu.pl
6. A. de Carvalho, M. Lyubich and M. Martens, Renormalization in the Hénon family, I: Universality but non-rigidity, J. Stat. Phys, 121 (2005), 611-669.
7. P. Collet, J.-P. Eckmann and H. Koch, Period doubling bifurcations for families of maps on Rn, J. Stat. Phys., 3D (1980).
8. P. Collet, J.-P. Eckmann and H. Koch, On universality for area-preserving maps of the plane, Physica D, 3 (1981), 457-467.
9. B. Derrida and Y. Pomeau, Feigenbaum's ratios of two-dimensional area-preserving maps, Phys. Lett. A, 80 (1980), 217-219.
10. P. Duarte, Persistent homoclinic tangencies for conservative maps near identity, Ergod. Th. & Dynam. Sys., 20 (2000), 393-438.
11. J.-P. Eckmann, H. Koch and P. Wittwer, Existence of a fixed point of the doubling transformation for area-preserving maps of the plane, Phys. Rev. A, 26 (1982), 720-722.
12. J.-P. Eckmann, H. Koch and P. Wittwer, A computer-assisted proof of universality for area-preserving maps, Memoirs of the American Mathematical Society, 47 (1984), vi+122 pp.
13. H. Epstein, New proofs of the existence of the Feigenbaum functions, Commun. Math. Phys., 106 (1986),395-426.
14. H. Epstein, Fixed points of composition operators II, Nonlinearity, 2 (1989), 305-310
15. D. F. Escande and F. Doveil, Renormalization method for computing the threshold of the large scale stochastic instability in two degree of freedom Hamiltonian systems, J.Stat. Phys., 26 (1981),257-284.
16. E. de Faria, Proof of Universality for Critical Circle Mappings, Thesis, CUNY, 1992
17. E. de Faria, Asymptotic rigidity of scaling ratios for critical circle mappings, Ergodic Theory Dynam. Systems, 19 (1999), 995-1035.
18. M. J. Feigenbaum, Quantitative universality for a class of nonlinear transformations, J. Stat. Phys., 19 (1978), 25-52.
19. M. J. Feigenbaum, The universal metric properties of nonlinear transformations, J. Stat. Phys., 21 (1979),669-706.
20. D. Gaidashev, Renormalization of isoenergetically degenerate Hamiltonian flows and associated bifurcations of invariant tori, Discrete Contin. Dyn. Syst., 13 (2005), 63-102.
21. D. Gaidashev, Cylinder renormalization for Siegel disks and a constructive measurable Riemann mapping theorem, Nonlinearity, 20 (2007), 713-742.
22. D. Gaidashev and H. Koch, Renormalization and shearless invariant tori: Numerical results, Nonlinearity, 17 (2004), 1713-1722.
23. D. Gaidashev and H. Koch, Period doubling in area-preserving maps: An associated one-dimenisonal problem, e-print arXiv:0811.2588 at Arxiv.org.
24. D. Gaidashev and M. Yampolsky, Cylinder renormalization of Siegel disks, Exp. Math., 16 (2007),215-226.
25. D. Gaidashev and T. Johnson, Dynamics of the universal area-preserving map associated with period doubling: Hyperbolic sets, Nonlinearity, 22 (2009), 2487-2520.
26. R. H. G. Helleman, Self-generated chaotic behavior in nonlinear mechanics, Fundamental problems in statistical mechanics, V (Proc. Fifth Internat. Summer School, Enschede, 1980), 165-233, North-Holland, Amsterdam-New York, 1980
27. A. Katok and B. Hasselblat, Introduction to the Modern Theory of Dynamical Systems, With a supplementary chapter by Katok and Leonardo Mendoza. Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995.
28. K. Khanin, J. Lopes Dias and J. Marklof, Multidimensional continued fractions, dynamic renormalization and KAM theory, Comm. Math. Phys., 270 (2007), 197-231.
29. H. Koch, On the renormalization of Hamiltonian flows, and critical invariant tori, Discrete Contin. Dyn. Syst., 8 (2002), 633-646.
30. H. Koch, A renormalization group fixed point associated with the breakup of golden invariant tori, Discrete Contin. Dyn. Syst., 11 (2004), 881-909.
31. H. Koch, Existence of critical invariant tori, Ergodic theory and dynamical systems, 28 (2008), 1879-1894
32. S. Kocić, Renormalization of Hamiltonians for Diophantine frequency vectors and KAM tori, Nonlinearity, 18 (2005), 2513-2544
33. H. Kokubu, D. Wilczak and P. Zgliczynski, Rigorous verification of cocoon bifurcations in the Michelson system, Nonlinearity, 20 (2007), 2147-2174
34. M. Lyubich, Feigenbaum-Coullet-Tresser universality and Milnor's hairness conjecture, Annals of Mathematics, 149 (1999), 319-420.
35. M. Lyubich and M. Martens, Renormalization in the Hénon family, II: The heteroclinic web, preprint, (2008), arXiv:0804.0780.
36. R. S. MacKay, Renormalisation in Area-Preserving Maps, Advanced Series in Nonlinear Dynamics, 6, World Scientific Publishing Co., Inc., River Edge, NJ, 1993.
37. R. S. MacKay, Renormalization approach to invariant circles in area-preserving maps, Physica D, 7 (1983), 283-300.
38. A. Mehr and D. F. Escande, Destruction of KAM tori in Hamiltonian systems: Link with the distabilization of nearby cycles and calculation of residues, Physica D, 13 (1984), 302-338.
39. W. de Melo and A. A. Pinto, Rigidity of C2 infinitely renormalizable unimodal maps, Comm. Math. Phys, 208 (1999), 91-105.
40. C. McMullen, Renormalization and 3-Manifolds Which Fiber over the Circle, (Annals of Mathematics Studies, 142), Princeton University Press, Princeton, NJ, 1996
41. C. McMullen, Self-similarity of Siegel disks and Hausdorff dimension of Julia sets, Acta Math., 180 (1998),247-292.
42. W. Pałuba, The Lipschitz condition for the conjugacies of Feigenbaum-like mappings, Fund. Math., 132 (1989), 227-258.
43. D. A. Rand, Global phase space universality, smooth conjugacies and renormalisation. I. The C1+case, Nonlinearity, 1 (1988), 181-202.
44. S. J. Shenker and L. P. Kadanoff, Critical behavior of KAM surfaces. I Empirical results, J. Stat. Phys., 27 (1982), 631-656.
45. D. Sullivan, Bounds, quadratic differentials and renormalization conjectures, AMS Centennial Publications, Vol. II, Amer. Math. Soc., Providence, R.I., (1992), 417-466
46. C. Tresser and P. Coullet, Itérations d'endomorphismes et groupe de renormalisation, (French) C. R. Acad. Sci. Paris Sér. A-B, 287 (1978), A577-A580
47. M. Yampolsky, Hyperbolicity of renormalization of critical circle maps, Publ. Math. Inst. Hautes Etudes Sci., 96 (2002), 1-41.
48. M. Yampolsky, Renormalization horseshoe for critical circle maps, Commun. Math. Physics, 240 (2003),75-96.
49. M. Yampolsky, Siegel disks and renormalization fixed points, Holomorphic Dynamics and Renormalization, 377-393, Fields Inst. Commun., vol 53, Amer. Math. Soc., Providence, RI, 2008.
50. P. Zgliczynski, Computer assisted proof of chaos in the Rössler equations and in the Hénon map, Nonlinearity, 10 (1997), 243-252.
51. P. Zgliczynski, Covering relations, cone conditions and the stable manifold theorem, Journal of Differential Equations, 246 (2009), 1774-1819
52. P. Zgliczynski and M. Gidea, Covering relations for multidimensional dynamical systems, J. Differential Equations, 202 (2004), 32-58
53. Programs available at http://www.math.uu.se/gaidash