Puzzle geometry and rigidity: The Fibonacci cycle is hyperbolic

Smania, Daniel

doi:10.1090/S0894-0347-07-00550-4

Puzzle geometry and rigidity: The Fibonacci cycle is hyperbolic

Author: Daniel Smania
Journal: J. Amer. Math. Soc. 20 (2007), 629-673
MSC (2000): Primary 37F30, 37C15, 30C62, 30C65, 37F25, 37F45, 37E20
DOI: https://doi.org/10.1090/S0894-0347-07-00550-4
Published electronically: January 17, 2007
MathSciNet review: 2291915
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Abstract: We describe a new and robust method to prove rigidity results in complex dynamics. The new ingredient is the geometry of the critical puzzle pieces: under control of geometry and ``complex bounds'', two generalized polynomial-like maps which admit a topological conjugacy, quasiconformal outside the filled-in Julia set, are indeed quasiconformally conjugate. The proof uses a new abstract removability-type result for quasiconformal maps, following ideas of Heinonen and Koskela and of Kallunki and Koskela, optimized for applications in complex dynamics. We prove, as the first application of this new method, that, for even criticalities distinct from two, the period two cycle of the Fibonacci renormalization operator is hyperbolic with -dimensional unstable manifold.

[AB] Lars Ahlfors and Lipman Bers, Riemann’s mapping theorem for variable metrics, Ann. of Math. (2) 72 (1960), 385–404. MR 115006, https://doi.org/10.2307/1970141
[ALdM] Artur Avila, Mikhail Lyubich, and Welington de Melo, Regular or stochastic dynamics in real analytic families of unimodal maps, Invent. Math. 154 (2003), no. 3, 451–550. MR 2018784, https://doi.org/10.1007/s00222-003-0307-6
[B] B. Bojarski, Remarks on Sobolev imbedding inequalities, Complex analysis, Joensuu 1987, Lecture Notes in Math., vol. 1351, Springer, Berlin, 1988, pp. 52–68. MR 982072, https://doi.org/10.1007/BFb0081242
[Br] Bodil Branner, Cubic polynomials: turning around the connectedness locus, Topological methods in modern mathematics (Stony Brook, NY, 1991) Publish or Perish, Houston, TX, 1993, pp. 391–427. MR 1215972
[BH98] Bodil Branner and John H. Hubbard, The iteration of cubic polynomials. I. The global topology of parameter space, Acta Math. 160 (1988), no. 3-4, 143–206. MR 945011, https://doi.org/10.1007/BF02392275
[Bru] Henk Bruin, Topological conditions for the existence of absorbing Cantor sets, Trans. Amer. Math. Soc. 350 (1998), no. 6, 2229–2263. MR 1458316, https://doi.org/10.1090/S0002-9947-98-02109-6
[BKNS] H. Bruin, G. Keller, T. Nowicki, and S. van Strien, Wild Cantor attractors exist, Ann. of Math. (2) 143 (1996), no. 1, 97–130. MR 1370759, https://doi.org/10.2307/2118654
[Bu] Xavier Buff, Fibonacci fixed point of renormalization, Ergodic Theory Dynam. Systems 20 (2000), no. 5, 1287–1317. MR 1786715, https://doi.org/10.1017/S0143385700000705
[GS96] Jacek Graczyk and Grzegorz Światek, Induced expansion for quadratic polynomials, Ann. Sci. École Norm. Sup. (4) 29 (1996), no. 4, 399–482 (English, with English and French summaries). MR 1386222
[GS97] Jacek Graczyk and Grzegorz Światek, Generic hyperbolicity in the logistic family, Ann. of Math. (2) 146 (1997), no. 1, 1–52. MR 1469316, https://doi.org/10.2307/2951831
[HK95] Juha Heinonen and Pekka Koskela, Definitions of quasiconformality, Invent. Math. 120 (1995), no. 1, 61–79. MR 1323982, https://doi.org/10.1007/BF01241122
[KK00] Sari Kallunki and Pekka Koskela, Exceptional sets for the definition of quasiconformality, Amer. J. Math. 122 (2000), no. 4, 735–743. MR 1771571
[KN] Gerhard Keller and Tomasz Nowicki, Fibonacci maps re(al)visited, Ergodic Theory Dynam. Systems 15 (1995), no. 1, 99–120. MR 1314971, https://doi.org/10.1017/S0143385700008269
[LV] O. Lehto and K. I. Virtanen, Quasiconformal mappings in the plane, 2nd ed., Springer-Verlag, New York-Heidelberg, 1973. Translated from the German by K. W. Lucas; Die Grundlehren der mathematischen Wissenschaften, Band 126. MR 0344463
[LvS] Genadi Levin and Sebastian van Strien, Local connectivity of the Julia set of real polynomials, Ann. of Math. (2) 147 (1998), no. 3, 471–541. MR 1637647, https://doi.org/10.2307/120958
[Lyu93] M. Lyubich.
Teichmuller space of Fibonacci maps.
Preprint, ims93-12, IMS-SUNY at Stony Brook, 1993.
[Lyu97] Mikhail Lyubich, Dynamics of quadratic polynomials. I, II, Acta Math. 178 (1997), no. 2, 185–247, 247–297. MR 1459261, https://doi.org/10.1007/BF02392694
[Lyu94] Mikhail Lyubich, Combinatorics, geometry and attractors of quasi-quadratic maps, Ann. of Math. (2) 140 (1994), no. 2, 347–404. MR 1298717, https://doi.org/10.2307/2118604
[Lyu99] Mikhail Lyubich, Feigenbaum-Coullet-Tresser universality and Milnor’s hairiness conjecture, Ann. of Math. (2) 149 (1999), no. 2, 319–420. MR 1689333, https://doi.org/10.2307/120968
[LM] Mikhail Lyubich and John Milnor, The Fibonacci unimodal map, J. Amer. Math. Soc. 6 (1993), no. 2, 425–457. MR 1182670, https://doi.org/10.1090/S0894-0347-1993-1182670-0
[dMvS] Welington de Melo and Sebastian van Strien, One-dimensional dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 25, Springer-Verlag, Berlin, 1993. MR 1239171
[McM] Curtis T. McMullen, Renormalization and 3-manifolds which fiber over the circle, Annals of Mathematics Studies, vol. 142, Princeton University Press, Princeton, NJ, 1996. MR 1401347
[P] R. A. Perez.
On the combinatorics of the principal nest.
Ph. D. Thesis, IMS-SUNY at Stony Brook, 2002.
[PR99] Feliks Przytycki and Steffen Rohde, Rigidity of holomorphic Collet-Eckmann repellers, Ark. Mat. 37 (1999), no. 2, 357–371. MR 1714763, https://doi.org/10.1007/BF02412220
[Sh] Weixiao Shen, On the metric properties of multimodal interval maps and 𝐶² density of Axiom A, Invent. Math. 156 (2004), no. 2, 301–403. MR 2052610, https://doi.org/10.1007/s00222-003-0343-2
[Sm01] Daniel Smania, Phase space universality for multimodal maps, Bull. Braz. Math. Soc. (N.S.) 36 (2005), no. 2, 225–274. MR 2152018, https://doi.org/10.1007/s00574-005-0038-y
[Sm02] Daniel Smania, On the hyperbolicity of the period-doubling fixed point, Trans. Amer. Math. Soc. 358 (2006), no. 4, 1827–1846. MR 2186998, https://doi.org/10.1090/S0002-9947-05-03803-1
[vSN94] S. van Strien and T. Nowicki.
Polynomial maps with a Julia set of positive Lebesgue measure: Fibonacci maps.
Unpublished, 1994.
[Sul92] Dennis Sullivan, Bounds, quadratic differentials, and renormalization conjectures, American Mathematical Society centennial publications, Vol. II (Providence, RI, 1988) Amer. Math. Soc., Providence, RI, 1992, pp. 417–466. MR 1184622
[SV] Grzegorz Światek and Edson Vargas, Decay of geometry in the cubic family, Ergodic Theory Dynam. Systems 18 (1998), no. 5, 1311–1329. MR 1653256, https://doi.org/10.1017/S0143385798117558