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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 $ 1$-dimensional unstable manifold.


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  • [AB] L. Ahlfors and L. Bers.
    Riemann's mapping theorem for variable metrics.
    Ann. of Math., 72:385-404, 1960. MR 0115006 (22:5813)
  • [ALdM] A. Avila, M. Lyubich and W. de Melo.
    Regular or stochastic dynamics in real analytic families of unimodal maps. Invent. Math., 154:451-550, 2003. MR 2018784 (2006i:37083)
  • [B] B. Bojarski.
    Remarks on Sobolev imbedding inequalities.
    In Proc. of the Conference on Complex Analysis, Joensuu, 1987, Lecture Notes in Math., 1351, Springer-Verlag, 1988. MR 0982072 (90b:46068)
  • [Br] B. Branner.
    Cubic polynomials: turning around the connectedness locus.
    In Topological methods in modern mathematics (Stony Brook, NY, 1991), Publish or Perish, Houston, 1993. MR 1215972 (94c:58168)
  • [BH98] B. Branner and J. Hubbard.
    The iteration of cubic polynomials. I. The global topology of parameter space.
    Acta Math., 160:143-206, 1998. MR 0945011 (90d:30073)
  • [Bru] H. Bruin.
    Topological conditions for the existence of absorbing Cantor sets.
    Trans. Amer. Math. Soc., 350:2229-2263, 1998. MR 1458316 (99f:58064)
  • [BKNS] H. Bruin, G. Keller, T. Nowicki and S. van Strien.
    Wild Cantor attractors exist.
    Ann. of Math. (2), 143:97-130, 1996. MR 1370759 (96m:58145)
  • [Bu] X. Buff.
    Fibonacci fixed point of renormalization.
    Ergodic Theory and Dynam. Systems, 20:1287-1317, 2000. MR 1786715 (2001m:37086)
  • [GS96] J. Graczyk and G. Swiatek.
    Induced expansion for quadratic polynomials.
    Ann. Sci. École Norm. Sup. (4), 29:399-482, 1996. MR 1386222 (98d:58152)
  • [GS97] J. Graczyk and G. Swiatek.
    Generic hyperbolicity in the logistic family.
    Ann. of Math. (2), 146:1-52, 1997. MR 1469316 (99b:58079)
  • [HK95] J. Heinonen and P. Koskela.
    Definitions of quasiconformality.
    Invent. Math., 120:61-79, 1995. MR 1323982 (96e:30051)
  • [KK00] S. Kallunki and P. Koskela.
    Exceptional sets for the definition of quasiconformality.
    American Journal of Mathematics, 122:735-743, 2000. MR 1771571 (2001h:37095)
  • [KN] G. Keller and T. Nowicki.
    Fibonacci maps re(al)visited.
    Ergodic Theory and Dynam. Systems, 15:99-120, 1995. MR 1314971 (95k:58098)
  • [LV] O. Lehto and K. Virtanen.
    Quasiconformal mappings in the plane.
    Springer-Verlag, New York-Heidelberg, 1973. MR 0344463 (49:9202)
  • [LvS] G. Levin and S. van Strien.
    Local connectivity of the Julia set of real polynomials.
    Ann. of Math., 147:471-541, 1998. MR 1637647 (99e:58143)
  • [Lyu93] M. Lyubich.
    Teichmuller space of Fibonacci maps.
    Preprint, ims93-12, IMS-SUNY at Stony Brook, 1993.
  • [Lyu97] M. Lyubich.
    Dynamics of quadratic polynomials. I, II.
    Acta Math, 178:185-297, 1997. MR 1459261 (98e:58145)
  • [Lyu94] M. Lyubich.
    Combinatorics, geometry and attractors of quasi-quadratic maps.
    Ann. of Math. (2), 140:347-404, 1994. MR 1298717 (95j:58108)
  • [Lyu99] M. Lyubich.
    Feigenbaum-Collet-Tresser universality and Milnor's hairness conjecture.
    Ann. of Math., 149:319-420, 1999. MR 1689333 (2000d:37051)
  • [LM] M. Lyubich and J. Milnor.
    The Fibonacci unimodal map.
    J. Am. Math. Soc., 6:425-457, 1993. MR 1182670 (93h:58080)
  • [dMvS] W. de Melo and S. van Strien.
    One-dimensional dynamics.
    Springer-Verlag, Berlin, 1993. MR 1239171 (95a:58035)
  • [McM] C. McMullen.
    Renormalization and $ 3$-manifolds which fiber over the circle.
    Annals of Mathematics Studies, 142, Princeton University Press, Princenton, 1996. MR 1401347 (97f:57022)
  • [P] R. A. Perez.
    On the combinatorics of the principal nest.
    Ph. D. Thesis, IMS-SUNY at Stony Brook, 2002.
  • [PR99] F. Przytycki and S. Rohde.
    Rigidity of holomorphic Collet-Eckmann repellers.
    Ark. Mat., 37:357-371, 1999. MR 1714763 (2000i:37064)
  • [Sh] W. Shen.
    On the metric property of multimodal maps and $ C^2$ density of Axiom A. Invent. Math., 156, no. 2, 301-403, 2004. MR 2052610 (2005d:37078)
  • [Sm01] D. Smania.
    Phase space universality for multimodal maps.
    Bull. Braz. Math. Soc, New Series 36(2), 225-274, 2005. MR 2152018 (2006f:37057)
  • [Sm02] D. Smania.
    On the hyperbolicity of the period-doubling fixed point.
    Trans. Amer. Math. Soc., 358, no. 4, 1827-1846, 2006. MR 2186998
  • [vSN94] S. van Strien and T. Nowicki.
    Polynomial maps with a Julia set of positive Lebesgue measure: Fibonacci maps.
    Unpublished, 1994.
  • [Sul92] D. Sullivan.
    Bounds, quadratic differentials, and renormalization conjectures.
    In AMS Centennial Publications: Mathematics into the Twenty-first Century, volume 2. AMS, 1992. MR 1184622 (93k:58194)
  • [SV] G. Swiatek and E. Vargas.
    Decay of geometry in the cubic family.
    Ergodic Theory Dynan. Systems, 18:1311-1329, 1998. MR 1653256 (99h:58161)

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Additional Information

Daniel Smania
Affiliation: Institute for Mathematical Sciences, State University of New York at Stony Brook, Stony Brook, New York 11794-3660
Address at time of publication: Departamento de Matemática, ICMC-USP-Campus de São Carlos, Caixa Postal 668, São Carlos-SP, CEP 13560-970, Brazil
Email: smania@icmc.usp.br

DOI: https://doi.org/10.1090/S0894-0347-07-00550-4
Keywords: Fibonacci combinatorics, generalized renormalization, puzzle, rigidity, quasiconformal conjugacy, removability, hyperbolicity
Received by editor(s): April 19, 2005
Published electronically: January 17, 2007
Additional Notes: This work was supported by grants CNPq-200764/01-2, CNPq-472316/2003-6 and FAPESP-03/03107-9. Visits to Warwick University and ICTP-Trieste were supported by CNPq-Brazil grant 460110/00-4, Warwick University and ICTP-Trieste.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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