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Renormalization in the golden-mean semi-Siegel Hénon family: universality and non-rigidity

Part of: Complex dynamical systems Holomorphic mappings and correspondences Low-dimensional dynamical systems

Published online by Cambridge University Press:  25 September 2018

JONGUK YANG
JONGUK YANG*
Affiliation:
Stony Brook University, Mathematics, 5 Old Field Road, East Setauket, NY 11733, USA email jongukyang@gmail.com
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Abstract

It was recently shown in Gaidashev and Yampolsky [Golden mean Siegel disk universality and renormalization. Preprint, 2016, arXiv:1604.00717] that appropriately defined renormalizations of a sufficiently dissipative golden-mean semi-Siegel Hénon map converge super-exponentially fast to a one-dimensional renormalization fixed point. In this paper, we show that the asymptotic two-dimensional form of these renormalizations is universal and is parameterized by the average Jacobian. This is similar to the limit behavior of period-doubling renormalizations in the Hénon family considered in de Carvalho et al [Renormalization in the Hénon family, I: universality but non-rigidity. J. Stat. Phys.121 (5/6) (2006), 611–669]. As an application of our result, we prove that the boundary of the golden-mean Siegel disk of a dissipative Hénon map is non-smoothly rigid.

Keywords

low-dimensional dynamicssmooth dynamicstiling

MSC classification

Primary: 37F25: Renormalization
Secondary: 37E20: Universality, renormalization 32H50: Iteration problems
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 4 , April 2020 , pp. 1108 - 1152
Copyright
© Cambridge University Press, 2018

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