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
DOI:
https://doi.org/10.1090/tran/6940
Published electronically:
September 15, 2017
Full-text PDF
Abstract | References | Similar Articles | Additional Information
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 . 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
or an initial segment of this
-tuple) increases with the number of periods. In comparison, among
-embeddings of the 2-disk with
, 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
Email:
pete@ime.usp.br
M. Martens
Affiliation:
Institute for Mathematical Sciences, Stony Brook University, Stony Brook, New York 11794
Email:
marco@math.stonybrook.edu
C. Tresser
Affiliation:
T. J. Watson Research Center, IBM, Yorktown Heights, New York 10598
DOI:
https://doi.org/10.1090/tran/6940
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