Hyperbolicity of renormalization of critical circle maps

Yampolsky, Michael

Publications Mathématiques de l'IHÉS, Tome 96 (2003), pp. 1-41.

[BR] L. Bers and H. L. Royden, Holomorphic families of injections, Acta Math., 157 (1986), 259-286. | MR | Zbl

[Do] A. Douady, Does a Julia set depend continuously on the polynomial?, in Complex dynamical systems: The mathematics behind the Mandelbrot set and Julia sets, R. L. Devaney (ed.), Proc. of Symposia in Applied Math., Vol. 49, Amer. Math. Soc., 1994, pp. 91-138. | MR | Zbl

[DH1] A. Douady and J. H. Hubbard, Etude dynamique des polynômes complexes, I-II, Pub. Math. d'Orsay, 1984. | Zbl

[DH2] A. Douady and J. H. Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. Éc. Norm. Sup., 18 (1985), 287-343. | EuDML | Numdam | MR | Zbl

[dF1] E. DE FARIA, Proof of universality for critical circle mappings, Thesis, CUNY, 1992.

[dF2] E. De Faria, Asymptotic rigidity of scaling ratios for critical circle mappings, Ergodic Theory Dynam. Systems, 19 (1999), no. 4, 995-1035. | MR | Zbl

[dFdM1] E. De Faria and W. De Melo, Rigidity of critical circle mappings I, J. Eur. Math. Soc. (JEMS), 1 (1999), no. 4, 339-392. | EuDML | MR | Zbl

[dFdM2] E. De Faria and W. De Melo, Rigidity of critical circle mappings II, J. Amer. Math. Soc., 13 (2000), no. 2, 343-370. | MR | Zbl

[Ep1] A. EPSTEIN, Towers of finite type complex analytic maps, PhD Thesis, CUNY, 1993.

[EKT] A. Epstein, L. Keen and C. Tresser, The set of maps $F_{a,b}:x\mapsto x+a+(b/2\pi )sin\left(2\pi x\right)$ with any given rotation interval is contractible, Commun. Math. Phys., 173 (1995), 313-333. | MR | Zbl

[EY] A. EPSTEIN and M. YAMPOLSKY, The universal parabolic map. Erg. Th. & Dyn. Systems, to appear.

[EE] J.-P. Eckmann and H. Epstein, On the existence of fixed points of the composition operator for circle maps, Commun. Math. Phys., 107 (1986), 213-231. | MR | Zbl

[FKS] M. FEIGENBAUM, L. KADANOFF, and S. SHENKER, Quasi-periodicity in dissipative systems. A renormalization group analysis, Physica, 5D (1982), 370-386. | MR

[He] M. HERMAN, Conjugaison quasi-symmetrique des homeomorphismes analytiques du cercle a des rotations, manuscript.

[Keen] L. Keen, Dynamics of holomorphic self-maps of C, in Holomorphic functions and moduli I, D. DRASIN et al. (eds.), Springer-Verlag, New York, 1988. | MR | Zbl

[Lan1] O. E. LANFORD, Renormalization group methods for critical circle mappings with general rotation number, in VIIIth International Congress on Mathematical Physics (Marseille, 1986), pp. 532-536, World Sci. Publishing, Singapore, 1987. | MR

[Lan2] O. E. Lanford, Renormalization group methods for critical circle mappings, Nonlinear evolution and chaotic phenomena, NATO Adv. Sci. Inst. Ser. B: Phys., 176, pp. 25-36, Plenum, New York, 1988. | Zbl

[Lyu2] M. Lyubich, Renormalization ideas in conformal dynamics, Cambridge Seminar “Current Developments in Math.”, May 1995, pp. 155-184, International Press, 1995, Cambridge, MA. | Zbl

[Lyu3] M. Lyubich, Dynamics of quadratic polynomials, I-II, Acta Math., 178 (1997), 185-297. | MR | Zbl

[Lyu4] M. Lyubich, Feigenbaum-Coullet-Tresser Universality and Milnor's Hairiness Conjecture, Ann. of Math. (2), 149 (1999), no. 2, 319-420. | Zbl

[Lyu5] M. Lyubich, Almost every real quadratic map is either regular or stochastic, Ann. of Math. (2), 156 (2002), no. 1, 1-78. | MR

[LY] M. Lyubich and M. Yampolsky, Dynamics of quadratic polynomials: complex bounds for real maps, Ann. l'Inst. Fourier 47, 4 (1997), 1219-1255. | Numdam | Zbl

[MP] R. S. Mackay and I. C. Percival, Universal small-scale structure near the boundary of Siegel disks of arbitrary rotation number, Physica, 26D (1987), 193-202. | MR | Zbl

[MSS] R. Mañé, P. Sad and D. Sullivan, On the dynamics of rational maps, Ann. Sci. Éc. Norm. Sup., 16 (1983), 193-217. | Numdam | MR | Zbl

[McM1] C. Mcmullen, Complex dynamics and renormalization, Annals of Math. Studies, v.135, Princeton Univ. Press, 1994. | MR | Zbl

[McM2] C. Mcmullen, Renormalization and 3-manifolds which fiber over the circle, Annals of Math. Studies, Princeton University Press, 1996. | MR | Zbl

[Mes] B. D. MESTEL, A computer assisted proof of universality for cubic critical maps of the circle with golden mean rotation number, PhD Thesis, University of Warwick, 1985.

[Mil] J. Milnor, Dynamics in one complex variable, Introductory lectures, Friedr. Vieweg & Sohn, Braunschweig, 1999. | MR | Zbl

[MvS] W. De Melo and S. Van Strien, One dimensional dynamics, Springer, 1993. | MR | Zbl

[ORSS] S. Ostlund, D. Rand, J. Sethna, and E. Siggia, Universal properties of the transition from quasi- periodicity to chaos in dissipative systems, Physica, 8D (1983), 303-342. | MR | Zbl

[Sul1] D. Sullivan, Quasiconformal homeomorphisms and dynamics, topology and geometry, Proc. ICM-86, Berkeley, v. II, 1216-1228. | MR | Zbl

[Sul2] D. Sullivan, Bounds, quadratic differentials, and renormalization conjectures, AMS Centennial Publications, 2, Mathematics into Twenty-first Century (1992). | MR | Zbl

[Sh] M. Shishikura, The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets, Ann. of Math. (2), 147 (1998), no. 2, 225-267. | MR | Zbl

[Sw1] G. Swiatek, Rational rotation numbers for maps of the circle, Commun. Math. Phys., 119 (1988), 109-128. | MR | Zbl

[Ya1] M. Yampolsky, Complex bounds for renormalization of critical circle maps, Erg. Th. & Dyn. Systems, 19 (1999), 227-257. | MR | Zbl

[Ya2] M. Yampolsky, The attractor of renormalization and rigidity of towers of critical circle maps, Commun. Math. Phys., 218 (2001), no. 3, 537-568. | MR | Zbl

[Ya3] M. YAMPOLSKY, The global horseshoe for the renormalization of critical circle maps, Preprint, 2002.

[Yoc] J.-C. Yoccoz, Il n'ya pas de contre-example de Denjoy analytique, C.R. Acad. Sci. Paris, 298 (1984) série I, 141-144. | Zbl