Boxing-in of a blender in a Hénon-like family

Abstract

IntroductionThe 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. In this study, we consider a Hénon-like family in ℝ3, which is the only explicitly given example of a system known to feature a blender in a certain range of a parameter (corresponding to an expansion or contraction rate). More specifically, as part of its hyperbolic set, this family has a pair of saddle fixed points with one-dimensional stable or unstable manifolds. When there is a blender, the closure of these manifolds cannot be avoided by one-dimensional curves coming from an appropriate direction. This property has been checked for the Hénon-like family by the method of computing extremely long pieces of global one-dimensional manifolds to determine the parameter range over which a blender exists.MethodsIn this study, we take the complimentary and local point of view of constructing an actual three-dimensional box (a parallelopiped) 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, as well as its respective local invariant manifold.ResultsThis constitutes a three-dimensional horseshoe that, in contrast to the idealized 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 characterize properties of the hyperbolic set intuitively.DiscussionIn particular, we trace relevant edges of sub-boxes as a function of the parameter to provide additional geometric insight into when the hyperbolic set may or may not be a blender.