Real polynomial diffeomorphisms with maximal entropy: II. Small Jacobian

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.