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The boundary of hyperbolicity for Hénon-like families

Published online by Cambridge University Press:  01 August 2008

YONGLUO CAO ,
STEFANO LUZZATTO  and
ISABEL RIOS
YONGLUO CAO
Affiliation:
Department of Mathematics, Suzhou University, 215006, Jiangsu, PR China Institute of Mathematics, Fudan University, Shanghai, 200433, PR China (email: ylcao@suda.edu.cn, yongluocao@yahoo.com)
STEFANO LUZZATTO
Affiliation:
Department of Mathematics, Imperial College, 180 Queen’s Gate, London SW7 2AZ, UK (email: Stefano.Luzzatto@imperial.ac.uk)
ISABEL RIOS
Affiliation:
Universidade Federal Fluminense, Niteroi, RJ, Brazil (email: rios@mat.uff.br)
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Abstract

We consider C2 Hénon-like families of diffeomorphisms of and study the boundary of the region of parameter values for which the non-wandering set is uniformly hyperbolic. Assuming sufficient dissipativity, we show that the loss of hyperbolicity is caused by a first homoclinic or heteroclinic tangency and that uniform hyperbolicity estimates hold uniformly in the parameter up to the bifurcation parameter and even, to some extent, at the bifurcation parameter.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 4 , August 2008 , pp. 1049 - 1080
Copyright
Copyright © 2008 Cambridge University Press

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