Boundary of the horseshoe locus for the Henon family
Boundary of the horseshoe locus for the Henon family
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Henon 家族马蹄形轨迹的边界
DOI：
10.1137/18m1174684
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发表时间：
2018
影响因子：
2.1
通讯作者：
H. Takahasi
中科院分区：
文献类型：
--
作者：
Z. Arai;Y. Ishii;H. Takahasi
本文的目的是研究 Hénon 族参数轨迹的几何性质，其中马蹄形的均匀双曲性被破坏。作为一个应用，我们获得了相应的非均匀双曲 Hénon 图的“零温度”平衡测量的变分特征。证明方法还得出参数空间中双曲马蹄轨迹的边界由两个单调部分组成，这证实了[Z. Arai 和 Y. Ishii，Comm。数学。物理学，361（2018），第343--414页]。这些结果的证明基于 Arai 和 Ishii 论文中开发的机制，该机制采用了 Hénon 族的动力学和参数空间的复杂化以及计算机辅助。
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.