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Hénon-like maps with arbitrary stationary combinatorics

Published online by Cambridge University Press:  09 March 2011

P. E. HAZARD
P. E. HAZARD*
Affiliation:
IME-USP, Rua do Matão 1010, Cidade Universitaria, São Paulo, SP, 05508-090, Brasil (email: [email protected])
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Abstract

We extend the renormalization operator introduced in [A. de Carvalho, M. Martens and M. Lyubich. Renormalization in the Hénon family, I: universality but non-rigidity. J. Stat. Phys.121(5/6) (2005), 611–669] from period-doubling Hénon-like maps to Hénon-like maps with arbitrary stationary combinatorics. We show that the renormalization picture also holds in this case if the maps are taken to be strongly dissipative. We study infinitely renormalizable maps F and show that they have an invariant Cantor set 𝒪 on which F acts like a p-adic adding machine for some p>1. We then show, as for the period-doubling case in the work of de Carvalho, Martens and Lyubich [Renormalization in the Hénon family, I: universality but non-rigidity. J. Stat. Phys.121(5/6) (2005), 611–669], that the sequence of renormalizations has a universal form, but that the invariant Cantor set 𝒪 is non-rigid. We also show that 𝒪 cannot possess a continuous invariant line field.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 5 , October 2011 , pp. 1391 - 1443
Copyright
Copyright © Cambridge University Press 2011

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References

[1]Birkhoff, G., Martens, M. and Tresser, C.. On the scaling structure for period doubling. Astérisque 286 (2003), 167–186.Google Scholar
[2]Davie, A. M.. Period doubling for C 2+ϵ mappings. Comm. Math. Phys. 176(2) (1996), 261–272.Google Scholar
[3]de Carvalho, A., Martens, M. and Lyubich, M.. Renormalization in the Hénon family, I: universality but non-rigidity. J. Stat. Phys. 121(5/6) (2005), 611–669.CrossRefGoogle Scholar
[4]de Melo, W. and Palis, J. Jr. Geometric Theory of Dynamical Systems. Springer, New York, 1982.Google Scholar
[5]de Melo, W. and van Strien, S.. One Dimensional Dynamics (Ergebnisse der Mathematik und ihrer Grenzgebiete, 3). Springer, Berlin, 1996.Google Scholar
[6]Eckmann, J.-P. and Wittwer, B.. Computer Methods and Borel Summability Applied to Feigenbaum’s Equation (Lecture Notes in Physics, 227). Springer, Berlin, 1985.CrossRefGoogle Scholar
[7]Feigenbaum, M. J.. Presentation functions, fixed points and a theory of scaling functions. J. Stat. Phys. 52(3/4) (1988), 527–569.Google Scholar
[8]Jiang, Y., Morita, T. and Sullivan, D.. Expanding direction of the period doubling operator. Comm. Math. Phys. 144(3) (1992), 509–520.CrossRefGoogle Scholar
[9]Lyubich, M.. Feigenbaum–Collet–Tresser universality and Milnor’s hairiness conjecture. Ann. of Math. (2) 149 (1999), 319–420.Google Scholar
[10]McMullen, C.. Renormalization and 3-manifolds which Fiber Over the Circle (Annals of Mathematical Studies, 142). Princeton University Press, Princeton, NJ, 1996.CrossRefGoogle Scholar
[11]Rand, D. A.. Global phase space universality, smooth conjugacies and renormalisation. I. The C 1+α case. Nonlinearity 1(1) (1988), 181–202.Google Scholar
[12]Sullivan, D.. Differentiable Structure on Fractal-like Sets, Determined by Intrinsic Scaling Functions on Dual Cantor Sets (Proceedings of Symposia in Pure Mathematics: The Mathematical Heritage of Hermann Weyl, 48). Ed. Wells, R. O. Jr. American Mathematical Society, Providence, RI, 1988, pp. 15–23.Google Scholar