A blender is a hyperbolic set with a stable or unstable invariant manifold that behaves as a geometric object of a dimension larger than that of the respective manifold itself. Blenders have been constructed in diffeomorphisms with a phase space of dimension at least three. We consider here the question of how one can identify, characterize and also visualize the underlying hyperbolic set of a given diffeomorphism to verify whether it actually is a blender or not. More specifically, we employ advanced numerical techniques for the computation of global manifolds to identify the hyperbolic set and its stable and unstable manifolds in an explicit Hénon-like family of three-dimensional diffeomorphisms. This allows to determine and illustrate whether the hyperbolic set is a blender; in particular, we consider as a distinguishing feature the self-similar structure of the intersection set of the respective global invariant manifold with a plane. By checking and illustrating a denseness property, we are able to identify a parameter range over which the hyperbolic set is a blender, and we discuss and illustrate how the blender disappears.

中文翻译：

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如何将双曲集识别为搅拌器

混合器是具有稳定或不稳定不变歧管的双曲线组，其表现为几何对象，其尺寸大于相应歧管本身的几何对象。混合器已经被构造为具有至少三个维度的相空间的亚纯态。我们在这里考虑一个问题，即如何识别，表征和可视化给定微分形的基本双曲集，以验证其是否实际上是一个混合器。更具体地说，我们采用先进的数值技术来计算整体流形，以识别双曲集及其在明确的类似于Hénon的三维微分族中的稳定流形和不稳定流形。这样就可以确定和说明双曲集是否是一个混合器。尤其是，我们将各个整体不变流形与平面的交集的自相似结构视为一个区别特征。通过检查和说明密度属性，我们能够确定双曲线集是搅拌器的参数范围，并讨论和说明搅拌器如何消失。