稠密分支树作为类 Hénon 和类 Lozi 吸引子的模型
Densely branching trees as models for Hénon-like and Lozi-like attractors
AI总结：
本文证明类 Hénon 与类 Lozi 吸引子可由具有稠密分支点的度量树自然扩张建模，从而统一刻画其拓扑与动力学，并排除更简单的一维模型。
摘要：
受 Crovisier 和 Pujals 近期关于平面温和耗散微分同胚工作的启发，我们证明了其奇怪吸引子上的类 Hénon 映射和类 Lozi 映射共轭于具有稠密分支点集的度量树上映射的自然扩张（也称逆极限上的移位同胚）。因此，这些树能很好地逼近吸引子的拓扑，而树上的映射也为动力学提供了良好模型。据我们所知，这是平面中首批规范的双参数吸引子族，保证存在这样一个一维局部连通模型，将这些吸引子的拓扑与动力学联系起来。对于 Hénon 映射，该结论适用于一个具有正 Lebesgue 测度的参数集，它推广了 Benedicks–Carleson 参数和 Wang–Young 参数集，并进一步阐明了 Barge 在1987年的结果；Barge 曾证明，存在某些参数值，使吸引子上的 Hénon 映射不是任何分支一维流形上映射的自然扩张。对于 Lozi 映射，该结果适用于 Misiurewicz 于1980年给出的一个开参数集。我们的结果可视为 Williams 于1967年的经典结果向非一致双曲世界的推广。我们还证明，不存在更简单的一维模型。
英文摘要：
Inspired by a recent work of Crovisier and Pujals on mildly dissipative diffeomorphisms of the plane, we show that Hénon-like and Lozi-like maps on their strange attractors are conjugate to natural extensions (a.k.a. shift homeomorphisms on inverse limits) of maps on metric trees with dense set of branch points. In consequence, these trees very well approximate the topology of the attractors, and the maps on them give good models of the dynamics. To the best of our knowledge, these are the first examples of canonical two-parameter families of attractors in the plane for which one is guaranteed such a 1-dimensional locally connected model tying together topology and dynamics of these attractors. For the Hénon maps this applies to a positive Lebesgue measure parameter set generalizing the Benedicks-Carleson parameters, the Wang-Young parameter set, and sheds more light onto the result of Barge from 1987, who showed that there exist parameter values for which Hénon maps on their attractors are not natural extensions of any maps on branched 1-manifolds. For the Lozi maps the result applies to an open set of parameters given by Misiurewicz in 1980. Our result can be seen as a generalization to the non-uniformly hyperbolic world of a classical result of Williams from 1967. We also show that no simpler 1-dimensional models exist.
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