Boundary of the horseshoe locus for the Hénon family

Abstract: The purpose of this article is to investigate geometric properties of the parameter locus of the H\'enon family where the uniform hyperbolicity of a horseshoe breaks down. As an application, we obtain a variational characterization of equilibrium measures "at temperature zero" for the corresponding non-uniformly hyperbolic H\'enon maps. The method of the proof also yields that the boundary of the hyperbolic horseshoe locus in the parameter space consists of two monotone pieces, which confirms a conjecture in [AI]. The proofs of these results are based on the machinery developed in [AI] which employs the complexification of both the dynamical and the parameter spaces of the H\'enon family together with computer assistance.