How to identify a hyperbolic set as a blender

Abstract

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.

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

“…Owing to the skew-product nature of Equation (1), for any ξ , the orthogonal projections of the global manifolds W s (p ± ) and W u (p ± ) of H onto the (x, y)-plane are the global manifolds W s (p ± h ) and W u (p ± h ) of the Hénon map h. In particular, the respective two-dimensional global manifolds, W u (p ± ) for ξ > 1 and W s (p ± ) for 0 < ξ < 1, are the direct products of R (the z-direction) with W u (p ± h ) and W s (p ± h ), respectively. In either case, for ξ sufficiently close to 1 the (transitive) hyperbolic set is a blender [1,2]. As mentioned in Hittmeyer et al [1,2], we use the definition of a blender from Díaz et al [3] and Bonatti et al [5] that says, colloquially speaking, that is a blender if its one-dimensional global manifold-W s ( ) for ξ > 1 and W u ( ) for 0 < ξ < 1-acts geometrically as a set of dimension two.…”

“…In either case, for ξ sufficiently close to 1 the (transitive) hyperbolic set is a blender [1,2]. As mentioned in Hittmeyer et al [1,2], we use the definition of a blender from Díaz et al [3] and Bonatti et al [5] that says, colloquially speaking, that is a blender if its one-dimensional global manifold-W s ( ) for ξ > 1 and W u ( ) for 0 < ξ < 1-acts geometrically as a set of dimension two. In more technical terms, the requirement is that there exists a C 1 -open set of curve segments in the three-dimensional phase space that each intersect the respective one-dimensional manifold locally near .…”

“…We are motivated here by the questions: how can one check when a blender exists in a given system, and what are the required geometric properties? According to the definition from Hittmeyer et al [1,2], the hyperbolic set of the map H is a blender if (when seen from an approriate direction) the one-dimensional manifold W s ( ) or W u ( ), respectively, looks like a surface-although it is a Cantor set of curves when viewed along the z-direction. We refer to this defining characteristic of a blender as the carpet property [1,2].…”

“…According to the definition from Hittmeyer et al [1,2], the hyperbolic set of the map H is a blender if (when seen from an approriate direction) the one-dimensional manifold W s ( ) or W u ( ), respectively, looks like a surface-although it is a Cantor set of curves when viewed along the z-direction. We refer to this defining characteristic of a blender as the carpet property [1,2]. Since the one-dimensional global manifolds, W s (p ± ) and W u (p ± ) of the fixed points p ± are dense in W s ( ) or W u ( ), respectively, the carpet property can be verified numerically for the family H by checking whether these one-dimensional global manifolds fill out an area in projection.…”

“…The initial box is defined in a parameter-dependent way, and this allows us to characterize properties of the hyperbolic set intuitively.Discussion: In 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. KEYWORDS non-uniform hyperbolicity, three-dimensional di eomorphism, Hénon-like map, generalized horseshoe construction, global invariant manifolds, carpet property which we introduced and studied in Hittmeyer et al [1,2]; see also Díaz et al [3] for a Hénon-like endomorphism that motivated it. To date, the family H defined on R 3 is the only known example of a diffeomorphism with a blender given in explicit form, specifically by Equation (1).…”