Renormalization Horseshoe for Critical Circle Maps

Communications in Mathematical Physics

Volumn 240, Issue 1-2, 2003, Pages 75-96

  Yampolsky, Michael ^a  

^a UNIVERSITY OF TORONTO   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0142208331     PISSN: 00103616     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00220-003-0891-8     Document Type: Article

References (36)

1. [Do] Douady, A.: Does a Julia set depend continuously on the polynomial? In: Complex dynamical systems: The mathematics behind the Mandelbrot set and Julia sets. Devaney, R.L. (ed), Proc. of Symposia in Applied Math., Vol 49, Providence RI: Amer. Math. Soc. 1994, pp. 91-138
2. [DH1] Douady, A., Hubbard, J.H.: Etude dynamique des polynômes complexes, I-II. Pub. Math. d'Orsay, 1984
3. [DH2] Douady, A., Hubbard, J.H.: On the dynamics of polynomial-like mappings. Ann. Sci. Éc. Norm. Sup. 18, 287-343 (1985)
4. [dF1] de Faria, E.: Proof of universality for critical circle mappings. Thesis, CUNY, 1992
5. [dF2] de Faria, E.: Asymptotic rigidity of scaling ratios for critical circle mappings. Ergodic Theory Dynam. Systems 19(4), 995-1035 (1999)
6. [dFdM1] de Faria E. de Melo, W.: Rigidity of critical circle mappingsI.J. Eur. Math. Soc. (JEMS) 1(4), 339-392 (1999)
7. [dFdM2] de Faria, E., de Melo, W.: Rigidity of critical circle mappings II. J. Am. Math. Soc. 13(2), 343-370 (2000)
8. [Ep1] Epstein, A.: Towers of finite type complex analytic maps. PhD Thesis, CUNY, 1993
9. a,b: x → x + a + b/2π sin (2πx) with any given rotation interval is contractible. Commun. Math. Phys. 173, 313-333 (1995)
10. [EY] Epstein, A., Yampolsky, M.: The universal parabolic map. To appear in Ergodic Theory
11. [EE] Eckmann, J.-P., Epstein, H.: On the existence of fixed points of the composition operator for circle maps. Commun. Math. Phys. 107, 213-231 (1986)
12. [FKS] Feigenbaum, M., Kadanoff, L., Shenker, S.: Quasi-periodicity in dissipative systems. A renormalization group analysis. Physica 5D, 370-386 (1982)
13. [He] Herman, M.: Conjugaison quasi-symmetrique des homeomorphismes analytiques du cercle a des rotations. Manuscript
14. [Keen] Keen, L.: Dynamics of holomorphic self-maps of ℂ*. In: Holomorphic functions and moduli I Drasin D. et al. (eds.), New York: Springer-Verlag, 1988
15. [Lan1] Lanford, O.E.: Renormalization group methods for critical circle mappings with general rotation number. In: VIIIth International Congress on Mathematical Physics (Marseille, 1986), Singapore: World Sci. Publishing, 1987, pp. 532-536
16. [Lan2] Lanford, O.E.: Renormalization group methods for critical circle mappings. Nonlinear evolution and chaotic phenomena. In: NATO Adv. Sci. Inst. Ser. B:Phys. 176, New York: Plenum, 1988, pp. 25-36
17. [Lyu2] Lyubich, M.: Renormalization ideas in conformal dynamics. In: Cambridge Seminar "Current Developments in Math.", May 1995. Cambridge, MA: International Press, 1995, pp. 155-184
18. [Lyu3] Lyubich, M.: Dynamics of quadratic polynomials, I-II. Acta Math. 178, 185-297 (1997)
19. [Lyu4] Lyubich, M.: Feigenbaum-Coullet-Tresser Universality and Milnor's Hairiness Conjecture. Ann. Math. (2) 149(2), 319-420 (1999)
20. [Lyu5] Lyubich, M.: Almost every real quadratic map is either regular or stochastic. Ann. Math.(2) 156(1), 1-78 (2002)
21. [LY] Lyubich, M., Yampolsky, M.: Dynamics of quadratic polynomials: Complex bounds for real maps. Ann. 1'Inst. Fourier 47(4), 1219-1255 (1997)
22. [McM1] McMullen, C.: Complex dynamics and renormalization. Ann. Math. Studies, V.135, Princeton, NJ: Princeton Univ. Press, 1994
23. [McM2] McMullen, C.: Renormalization and 3-manifolds which fiber over the circle. Ann. Math. Studies, Princeton, NJ: Princeton University Press, 1996
24. [Mes] Mestel, B.D.: A computer assisted proof of universality for cubic critical maps of the circle with golden mean rotation number. PhD. Thesis, University of Warwick, 1985
25. [Mil] Milnor, J.: Dynamics in one complex variable. Introductory lectures, Braunschweig: Friedr. Vieweg & Sohn, 1999
26. [MvS] de Melo, W., van Strien, S.: One dimensional dynamics. Berlin-Heidelberg-New York: Springer-Verlag, 1993
27. [ORSS] Ostlund, S., Rand, D., Sethna, J., Siggia, E.: Universal properties of the transition from quasi-periodicity to chaos in dissipative systems. Physica 8D, 303-342 (1983)
28. [Sul1] Sullivan, D.: Quasiconformal homeomorphisms and dynamics, topology and geometry. Proc. ICM-86, Berkeley, V.II, pp. 1216-1228
29. [Sul2] D.Sullivan. Bounds, quadratic differentials, and renormalization conjectures. AMS Centennial Publications. 2: Mathematics into Twenty-first Century (1992).
30. [Sh] Shishikura, M.: The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets. Ann. Math. (2) 147(2), 225-267 (1998)
31. [Sw] Światek, G.: Rational rotation numbers for maps of the circle. Commun. Math. Phys. 119, 109-128 (1988)
32. [Ya1] Yampolsky, M.: Complex bounds for renormalization of critical circle maps. Erg. Th. & Dyn. Systems 19, 227-257 (1999)
33. [Ya2] Yampolsky, M.: The attractor of renormalization and rigidity of towers of critical circle maps. Commun. Math. Phys. 218(3), 537-568 (2001)
34. [Ya3] Yampolsky, M.: Hyperbolicity of renormalization of critical circle maps. Publ. Math IHÉS, to appear
35. [Ya4] Yampolsky, M.: On the eigenvalues of a renormalization operator. To appear in Nonlinearity
36. [Yoc] Yoccoz, J.-C.: Il n'ya pas de centre-example de Denjoy analytique. C.R. Acad. Sci. Paris 298 série I, 141-144 (1984)