How to identify a hyperbolic set as a blender (Q2211127)

The concept of \textit{blenders} first appeared in [\textit{C. Bonatti} and \textit{L. J. Díaz}, Ann. Math. (2) 143, No. 2, 357--396 (1996; Zbl 0852.58066)]. It arose as an example of robust nonuniformly hyperbolic systems, and is defined as a (transitive) hyperbolic set of a diffeomorphism at least three-dimensional for which its stable manifold acts geometrically as a set of higher dimension. For a diffeomorphism in a three-dimensional phase space, a hyperbolic set with unstable index 2 is a blender if, roughly speaking, its stable manifold acts geometrically like a surface (a property that is called \textit{the carpet property}). In this work, the authors consider the Hénon-like family \[ H(x,y,z) = (y,-9.5+y^2+0.3 x,\xi z+y), \] where \(\xi >0\) is a parameter that determines the attraction or repulsion of the \(z\)-coordinate. For this family, using numerical techniques, they compute the global manifolds in order to identify the hyperbolic set and its stable and unstable manifolds, and use this to verify if the hyperbolic set is, in fact, a blender. Making use of a denseness property (the intersection set of the global invariant manifold with a plane), they are able to identify a range of the parameter \(\xi\) for which the hyperbolic set is a blender. Finally, they present a discussion about the disappearance of blenders