Boundary of the horseshoe locus for the H\'enon family

TL;DR

This paper investigates the boundary of the hyperbolic horseshoe locus in the Hénon family, providing a variational characterization of equilibrium measures and confirming a conjecture about the boundary's structure using complexification and computer-assisted proofs.

Contribution

It offers a detailed geometric analysis of the horseshoe locus boundary, confirming a conjecture and developing a variational approach for equilibrium measures in non-uniform hyperbolic Hénon maps.

Findings

Boundary of the hyperbolic horseshoe locus consists of two monotone pieces

Provides a variational characterization of equilibrium measures at zero temperature

Confirms a conjecture about the boundary's structure in parameter space

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.