In this paper, we will assume M M to be a compact smooth manifold and f : M → M f:M\to M to be a diffeomorphism. We herein demonstrate that a C 1 {C}^{1} generic diffeomorphism f f is Axiom A and has no cycles if f f is asymptotic measure expansive. Additionally, for a C 1 {C}^{1} generic diffeomorphism f f , if a homoclinic class H ( p , f ) H\left(\hspace{0.08em}p,f) that contains a hyperbolic periodic point p p of f f is asymptotic measure-expansive, then H ( p , f ) H\left(\hspace{0.08em}p,f) is hyperbolic of f f .

中文翻译：

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泛微分同胚的渐近测度扩展性

在本文中，我们将假设 MM 是一个紧凑的光滑流形，而 f : M → M f:M\to M 是一个微分同胚。我们在此证明了 C 1 {C}^{1} 泛型微分同胚 ff 是公理 A 并且如果 ff 是渐近测度扩张的，则没有循环。此外，对于 C 1 {C}^{1} 泛型微分同胚 ff ，如果同宿类 H ( p , f ) H\left(\hspace{0.08em}p,f) 包含 ff 的双曲周期点 pp是渐近测度扩展的，则 H ( p , f ) H\left(\hspace{0.08em}p,f) 是 ff 的双曲线。