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Infinitely many moduli of stability at the dissipative boundary of chaos

Authors: P. Hazard, M. Martens and C. Tresser
Journal: Trans. Amer. Math. Soc. 370 (2018), 27-51
MSC (2010): Primary 37B40, 37C15, 37F25
Published electronically: September 15, 2017
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Abstract: In the family of area-contracting Hénon-like maps with zero topological entropy we show that there are maps with infinitely many moduli of stability. Thus one cannot find all the possible topological types for non-chaotic area-contracting Hénon-like maps in a family with finitely many parameters. A similar result, but for the chaotic maps in the family, became part of the folklore a short time after Hénon used such maps to produce what was soon conjectured to be the first non-hyperbolic strange attractor in $ \mathbb{R}^2$. Our proof uses recent results about infinitely renormalisable area-contracting Hénon-like maps; it suggests that the number of parameters needed to represent all possible topological types for area-contracting Hénon-like maps whose sets of periods of their periodic orbits are finite (and in particular are equal to $ \{1,\, 2,\dots ,\,2^{n-1}\}$ or an initial segment of this $ n$-tuple) increases with the number of periods. In comparison, among $ C^k$-embeddings of the 2-disk with $ k\geq 1$, the maximal moduli number for non-chaotic but non-area-contracting maps in the interior of the set of zero-entropy is infinite.

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Additional Information

P. Hazard
Affiliation: Department of Mathematics, Stony Brook University, Stony Brook, New York 11794
Address at time of publication: IME-USP, Rua do Matão, 1010 Cidade Universitária, São Paulo, SP Brazil – CEP 05508-090

M. Martens
Affiliation: Institute for Mathematical Sciences, Stony Brook University, Stony Brook, New York 11794

C. Tresser
Affiliation: T. J. Watson Research Center, IBM, Yorktown Heights, New York 10598

Received by editor(s): October 16, 2014
Received by editor(s) in revised form: November 23, 2015
Published electronically: September 15, 2017
Additional Notes: This work was partially supported by CNPq PVE Grant #401020/2014-2, NSF grant DMS-1600554, FAPESP grant 2008/10659-1, and Leverhulme Trust grant RPG-279.
Article copyright: © Copyright 2017 American Mathematical Society

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