no. 4
Dynamics of quadratic polynomials: complex bounds for real maps
Annales de l'Institut Fourier, Tome 47 (1997) no. 4, pp. 1219-1255.

Nous démontrons des bornes complexes pour les applications quadratiques réelles infiniment renormalisables dont la combinatoire est essentiellement bornée. C’est le dernier ingrédient manquant dans le problème des bornes complexes pour les applications quadratiques réelles infiniment renormalisables. Un de des corollaires est que l’ensemble de Julia de toute application quadratique réelle z↦z 2 +c,c∈[-2,1/4] est localement connexe.

We prove complex bounds for infinitely renormalizable real quadratic maps with essentially bounded combinatorics. This is the last missing ingredient in the problem of complex bounds for all infinitely renormalizable real quadratics. One of the corollaries is that the Julia set of any real quadratic map z↦z 2 +c, c∈[-2,1/4], is locally connected.

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Lyubich, Mikhail; Yampolsky, Michael. Dynamics of quadratic polynomials: complex bounds for real maps. Annales de l'Institut Fourier, Tome 47 (1997) no. 4, pp. 1219-1255. doi : 10.5802/aif.1598. http://www.numdam.org/articles/10.5802/aif.1598/

[BL1] A. Blokh and M. Lyubich, Measure of solenoidal attractors of unimodal maps of the segment., Math. Notes, 48 (1990), 1085-1990. | MR | Zbl

[BL2] A. Blokh and M. Lyubich, Measure and dimension of solenoidal attractors of one dimensional dynamical systems, Commun. of Math. Phys., 1 (1990), 573-583. | MR | Zbl

[D] A. Douady, Chirurgie sur les applications holomorphes, in Proc. ICM, Berkeley, 1986, 724-738. | MR | Zbl

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

[E] H. Epstein, Fixed points of composition operators, Nonlinearity, 2 (1989), 305-310. | MR | Zbl

[F] E. De Faria, Proof of universality for critical circle mappings, Thesis, CUNY (1992).

[G] J. Guckenheimer, Limit sets of S-unimodal maps with zero entropy, Comm. Math. Phys., 110 (1987), 655-659. | MR | Zbl

[GJ] J. Guckenheimer, S. Johnson, Distortion of S-unimodal maps, Ann. Math., 132 (1990), 71-130. | MR | Zbl

[GS1] J. Graczyk, G. Swiatek, Induced expansion for quadratic polynomials, Ann. Sci. Éc. Norm. Sup., 29 (1996), 399-482. | Numdam | MR | Zbl

[GS2] J. Graczyk, G. Swiatek, Polynomial-like property for real quadratic polynomials, Preprint, 1995. | Zbl

[H] J.H. Hubbard, Local connectivity of Julia sets and bifurcation loci: three theorems of J.-C. Yoccoz, in Topological Methods in Modern Mathematics, A Symposium in Honor of John Milnor's 60th Birthday, Publish or Perish, 1993. | Zbl

[HJ] J. Hu, Y. Jiang, The Julia set of the Feigenbaum quadratic polynomial is locally connected, Preprint, 1993.

[He] M. Herman, Conjugaison quasi-symétrique des homomorphismes analytiques du cercle à des rotations, Preprint.

[J] Y. Jiang, Infinitely renormalizable quadratic Julia sets, Preprint 1993. | Zbl

[LS] G. Levin, S. Van Strien, Local connectivity of Julia sets of real polynomials, Preprint IMS at Stony Brook, # 1995/5. | Zbl

[L1] M. Lyubich, Combinatorics, geometry and attractors of quasi-quadratic maps, Ann. Math, 140 (1994), 347-404. | Zbl

[L2] M. Lyubich, Dynamics of quadratic polynomials, I-II, Acta Math, 178 (1997), 185-297. | MR | Zbl

[M1] J. Milnor, Local connectivity of Julia sets: expository lectures, Preprint IMS at Stony Brook #1992/11.

[M2] J. Milnor, Periodic orbits, external rays and the Mandelbrot set: An expository account, Preprint, 1995.

[MT] J. Milnor, W. Thurston, On iterated maps of the interval, Lecture Notes in Mathematics 1342, Springer Verlag (1988), 465-563. | MR | Zbl

[McM1] C. Mcmullen, Complex dynamics and renormalization, Annals of Math. Studies, 135, Princeton University Press (1994). | Zbl

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

[Ma] M. Martens, Distortion results and invariant Cantor sets for unimodal maps, Erg. Th. and Dyn. Syst., 14 (1994), 331-349. | MR | Zbl

[MS] W. De Melo, S. Van Strien, One dimensional dynamics, Springer-Verlag (1993). | MR | Zbl

[R] M. Rees, A possible approach to a complex renormalization problem, in Linear and Complex Analysis Problem Book 3, part II, pp. 437-440, Lecture Notes in Math., 1574.

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

[Y] M. Yampolsky, Complex bounds for critical circle maps, Preprint IMS at Stony Brook, # 1995/12.

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