Boxing-in of a blender in a Hénon-like family
Stefanie Hittmeyer, Bernd Krauskopf, Hinke M. Osinga, and Katsutoshi Shinohara
Abstract
The extension of the Smale horseshoe construction for diffeomorphisms in the plane to those in spaces of at least dimension three may result in a hyperbolic invariant set referred to as a blender. The defining property of a blender is that it has a stable or unstable invariant manifold that appears to have a dimension larger than expected. We consider here a Hénon-like family in R3, which features a blender in a certain range of a parameter (corresponding to an expansion or contraction rate). We consider the one-dimensional stable or unstable manifolds of a pair of saddle fixed points: the hyperbolic set containing these saddle points is a blender when the closure of these manifolds, seen from an appropriate direction, cannot be avoided by one-dimensional curves. This property has been used to detect numerically over which parameter range a blender exists in the Hénon-like family. We take here the complimentary approach of constructing an actual three-dimensional box (a parallelepiped) that acts as an outer cover of the hyperbolic set. The successive forward or backward images of this box form a nested sequence of sub-boxes that contains the hyperbolic set and its respective local invariant manifold. This constitutes a three-dimensional horseshoe that, in contrast to the idealised affine construction, is quite general and features sub-boxes with curved edges. The initial box is defined in a parameter-dependent way, and this allows us to characterise the hyperbolic set further in an intuitive way. In particular, tracing relevant edges of sub-boxes as a function of the parameter provides additional geometric insight into when the hyperbolic set may or may not be a blender.
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