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arXiv · 1803.09381

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

Zin Arai·Yutaka Ishii·Hiroki Takahasi
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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.

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Zin Arai, Yutaka Ishii, Hiroki Takahasi. 2018-03-26. Boundary of the horseshoe locus for the H\'enon family. https://arxiv.org/abs/1803.09381

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