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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: pete@ime.usp.br)
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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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