A diffeomorphism of the plane is Anosov if it has a hyperbolic splitting at every point of the plane. In addition to linear hyperbolic automorphisms, translations of the plane also carry an Anosov structure (the existence of Anosov structures for plane translations was originally shown by White). Mendes conjectured that these are the only topological conjugacy classes for Anosov diffeomorphisms in the plane. Very recently, Matsumoto gave an example of an Anosov diffeomorphism of the plane, which is a Brouwer translation but not topologically conjugate to a translation, disproving Mendes’ conjecture. In this paper we prove that Mendes’ claim holds when the Anosov diffeomorphism is the time-one map of a flow, via a theorem about foliations invariant under a time-one map. In particular, this shows that the kind of counterexample constructed by Matsumoto cannot be obtained from a flow on the plane.

中文翻译：

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叶子和共轭，II：门德斯猜想的时间流图

平面的微分同胚是阿诺索夫如果它在平面的每个点都有一个双曲线分裂。除了线性双曲自同构之外，平面的平移也带有 Anosov 结构（平面平移的 Anosov 结构的存在最初由 White 证明）。Mendes 推测这些是平面上 Anosov 微分同胚的唯一拓扑共轭类。最近，Matsumoto 给出了一个平面的 Anosov 微分同胚的例子，这是一个 Brouwer 平移但不是拓扑共轭的平移，反驳了 Mendes 的猜想。在本文中，我们通过关于在时间一映射下叶面不变的定理证明当阿诺索夫微分同胚是流的一时间映射时，门德斯的主张成立。特别是，