We consider perturbations of normally hyperbolic invariant manifolds, under which they can lose their hyperbolic properties. We show that if the perturbed map which drives the dynamical system exhibits some topological properties, then the manifold is perturbed to an invariant set. The main feature is that our results do not require the rate conditions to hold after the perturbation. In this case the manifold can be perturbed to an invariant set, which is not a topological manifold. Our method is not perturbative. It can be applied to establish invariant sets within a prescribed neighbourhood also in the absence of a normally hyperbolic invariant manifold prior to perturbation. The work is in the setting of nonorientable Banach vector bundles, without needing to assume invertibility of the map.

中文翻译：

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在没有速率条件的情况下正常双曲不变流形的持久性

我们考虑通常双曲不变流形的扰动，在这种情况下，它们可能会失去双曲特性。我们表明，如果驱动动力系统的扰动图表现出一些拓扑特性，那么流形就会被扰动到一个不变集。主要特点是我们的结果不需要在扰动后保持速率条件。在这种情况下，流形可以被扰乱为一个不变集，它不是一个拓扑流形。我们的方法不是扰动的。在扰动之前没有正常双曲不变流形的情况下，它也可以应用于在规定的邻域内建立不变集。这项工作是在不可定向的 Banach 向量丛的设置中进行的，不需要假设地图的可逆性。