The purpose of this article is to investigate geometric properties of the parameter locus of the Hénon 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 nonuniformly hyperbolic Hénon 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 [Z. Arai and Y. Ishii, Comm. Math. Phys., 361 (2018), pp. 343-414]. The proofs of these results are based on the machinery developed in the paper by Arai and Ishii which employs the complexification of both the dynamical and parameter spaces of the Hénon family together with computer assistance.
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