Abstract Let X be a C1 vector field of a compact connected boundaryless smooth smooth Riemannian manifold M with dimM ≥ 3. We show the following: (i) if the chain recurrence set 𝓡(X) is robustly measure expansive, then it is Axiom A without cycles; and (ii) if the homoclinic class H(Orb(p), X) that contains a hyperbolic periodic orbit Orb(p) is generic-robustly measure expansive, then it is hyperbolic.

中文翻译：

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稳健地测量 C1 矢量场的扩展性

摘要 令 X 是一个紧致连通无边界光滑黎曼流形 M 的 C1 向量场，dimM ≥ 3。我们证明如下： (i) 如果链递推集 𝓡(X) 是稳健测度膨胀的，则它是公理 A无循环；(ii) 如果包含双曲周期轨道 Orb(p) 的同宿类 H(Orb(p), X) 是通用稳健测量膨胀的，则它是双曲线的。