The full renormalization horseshoe for unimodal maps of higher degree: exponential contraction along hybrid classes

Avila, Artur; Lyubich, Mikhail

Publications Mathématiques de l'IHÉS, Tome 114 (2011), pp. 171-223.

We prove exponential contraction of renormalization along hybrid classes of infinitely renormalizable unimodal maps (with arbitrary combinatorics), in any even degree d. We then conclude that orbits of renormalization are asymptotic to the full renormalization horseshoe, which we construct. Our argument for exponential contraction is based on a precompactness property of the renormalization operator (“beau bounds”), which is leveraged in the abstract analysis of holomorphic iteration. Besides greater generality, it yields a unified approach to all combinatorics and degrees: there is no need to account for the varied geometric details of the dynamics, which were the typical source of contraction in previous restricted proofs.

[AKLS] Avila, A.; Kahn, J.; Lyubich, M.; Shen, W. Combinatorial rigidity for unicritical polynomials, Ann. Math., Volume 170 (2009), pp. 783-797 | DOI | MR | Zbl

[ALM] Avila, A.; Lyubich, M.; Melo, W. Regular or stochastic dynamics in real analytic families of unimodal maps, Invent. Math., Volume 154 (2003), pp. 451-550 | DOI | MR | Zbl

[ALS] Avila, A.; Lyubich, M.; Shen, W. Parapuzzle of the Multibrot set and typical dynamics of unimodal maps, J. Eur. Math. Soc., Volume 13 (2011), pp. 27-56 | DOI | MR | Zbl

[Ca] Cartan, H. Sur les rétractions d’une variété, C. R. Acad. Sci. Paris Sér. I, Math., Volume 303 (1986), p. 715 | MR | Zbl

[Ch] Cheraghi, D. Combinatorial rigidity for some infinitely renormalizable unicritical polynomials, Conform. Geom. Dyn., Volume 14 (2010), pp. 219-255 | DOI | MR | Zbl

[Cv] Cvitanović, P. Universality in Chaos, Adam Hilger, Bristol, 1984 | Zbl

[D] Douady, A. Chirurgie sur les applications holomorphes, Proceedings of ICM-86, AMS, Providence (1987), pp. 724-738 | Zbl

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

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

[F] Feigenbaum, M. J. Quantitative universality for a class of non-linear transformations, J. Stat. Phys., Volume 19 (1978), pp. 25-52 | DOI | MR | Zbl

[FM] Faria, E.; Melo, W. Rigidity of critical circle mappings I, J. Eur. Math. Soc., Volume 1 (1999), pp. 339-392 | DOI | Zbl

[GS] Graczyk, J.; Swiatek, G. Generic hyperbolicity in the logistic family, Ann. Math. (2), Volume 146 (1997), pp. 1-52 | DOI | MR | Zbl

[Hi] Hinkle, B. Parabolic limits of renormalization, Ergod. Theory Dyn. Syst., Volume 20 (2000), pp. 173-229 | DOI | MR | Zbl

[K] J. Kahn, A priori bounds for some infinitely renormalizable quadratics: I. Bounded primitive combinatorics. Preprint IMS at Stony Brook, # 5 (2006).

[KL1] Kahn, J.; Lyubich, M. A priori bounds for some infinitely renormalizable quadratics: II. Decorations, Ann. Sci. Ecole Norm. Super., Volume 41 (2008), pp. 57-84 | Numdam | MR | Zbl

[KL2] Kahn, J.; Lyubich, M. A priori bounds for some infinitely renormalizable quadratics, III. Molecules, Complex Dynamics: Families and Friends. Proceeding of the conference dedicated to Hubbard’s 60th birthday, AK Peters, Wellesley (2009) | Zbl

[KSS] Kozlovski, O.; Shen, W.; Strien, S. Rigidity for real polynomials, Ann. Math., Volume 165 (2007), pp. 749-841 | DOI | Zbl

[KS] Krzyzewski, K.; Szlenk, W. On invariant measures for expanding differential mappings, Studia Math., Volume 33 (1969), pp. 83-92 | MR | Zbl

[La] Lanford, O. E. A computer assisted proof of the Feigenbaum conjectures, Bull. Am. Math. Soc., Volume 6 (1982), pp. 427-434 | DOI | MR | Zbl

[LvS] Levin, G.; Strien, S. Local connectivity of Julia sets of real polynomials, Ann. Math., Volume 147 (1998), pp. 471-541 | DOI | Zbl

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

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

[L3] Lyubich, M. Feigenbaum-Coullet-Tresser universality and Milnor’s hairiness conjecture, Ann. Math.  (2), Volume 149 (1999), pp. 319-420 | DOI | MR | Zbl

[L4] Lyubich, M. Almost every real quadratic map is either regular or stochastic, Ann. Math. (2), Volume 156 (2002), pp. 1-78 | DOI | MR | Zbl

[LY] Lyubich, M.; Yampolsky, M. Dynamics of quadratic polynomials: complex bounds for real maps, Ann. Inst. Fourier, Volume 47 (1997), pp. 1219-1255 | DOI | Numdam | MR | Zbl

[Lju1] Ljubich, Y. I. Introduction to the Theory of Banach Representations of Groups, Birkhäuser, Basel, 1988 | DOI | Zbl

[Lju2] Ljubich, Y. I. Dissipative actions and almost periodic representations of abelian semigroups, Ukr. Math. J., Volume 40 (1988), pp. 58-62 | DOI | Zbl

[Ma1] Martens, M. Distortion results and invariant Cantor sets for unimodal maps, Ergod. Theory Dyn. Syst., Volume 14 (1994), pp. 331-349 | DOI | MR | Zbl

[Ma2] Martens, M. The periodic points of renormalization, Ann. Math., Volume 147 (1998), pp. 543-584 | DOI | MR | Zbl

[McM1] McMullen, C. Complex Dynamics and Renormalization, Annals of Math. Studies, 135, Princeton University Press, Princeton, 1994 | Zbl

[McM2] McMullen, C. Renormalization and 3-Manifolds which Fiber over the Circle, Annals of Math. Studies, 135, Princeton University Press, Princeton, 1996 | Zbl

[MvS] Melo, W.; Strien, S. One-Dimensional Dynamics, Springer, Berlin, 1993 | Zbl

[Mi] Milnor, J. Periodic orbits, external rays, and the Mandelbrot set: expository lectures, Géometrie complexe et systémes dynamiques, Volume in Honor of Douady’s 60th Birthday (Astérisque, 261) (2000), pp. 277-333 | Numdam | Zbl

[MSS] Mañé, R.; Sad, P.; Sullivan, D. On the dynamics of rational maps, Ann. Sci. Ecole Norm. Super., Volume 16 (1983), pp. 193-217 | Numdam | Zbl

[Sl] Slodkowski, Z. Holomorphic motions and polynomial hulls, Proc. Am. Math. Soc., Volume 111 (1991), pp. 347-355 | DOI | MR | Zbl

[S] Sullivan, D. Bounds, Quadratic Differentials, and Renormalization Conjectures, AMS Centennial Publications, 2, 1992 (Mathematics into Twenty-first Century) | Zbl

[TC] Tresser, C.; Coullet, P. Itération d’endomorphismes et groupe de renormalisation, C. R. Acad. Sci. Paris A, Volume 287 (1978), pp. 577-580 | MR | Zbl