Boundary of the horseshoe locus for the Hénon family

Zin Arai, Yutaka Ishii, Hiroki Takahasi

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    2 Citations (Scopus)

    Abstract

    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.

    Original languageEnglish
    Pages (from-to)2234-2248
    Number of pages15
    JournalSIAM Journal on Applied Dynamical Systems
    Volume17
    Issue number3
    DOIs
    Publication statusPublished - 2018

    All Science Journal Classification (ASJC) codes

    • Analysis
    • Modelling and Simulation

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