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Real polynomial diffeomorphisms with maximal entropy: II. Small Jacobian

Published online by Cambridge University Press:  11 September 2006

ERIC BEDFORD
Affiliation:
Indiana University, Bloomington, IN 47405, USA (e-mail: bedford@indiana.edu)
JOHN SMILLIE
Affiliation:
Cornell University, Ithaca, NY 14853, USA (e-mail: smillie@math.cornell.edu)

Abstract

Let $\{h_{a,b} : a,b \in \mathbb{R}, b \neq 0\}$ denote the Hénon family of quadratic polynomial diffeomorphisms of $\mathbb{R}^2$, with $b$ equal to the Jacobian of $h_{a,b}$. In this paper, we describe the locus of parameter values $(a,b)$ such that $0 < |b| < 0.06$, and the restriction of $h_{a,b}$ to its non-wandering set is topologically conjugate to the horseshoe map. The boundary of the horseshoe locus is shown to be characterized by a homoclinic tangency which is part of a generic unfolding.

Type
Research Article
Copyright
2006 Cambridge University Press

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