Blenders in centre unstable Hénon-like families: with an application to heterodimensional bifurcations

Abstract

Abstract. We give an explicit family of polynomial maps called center unstable Hénon-like maps and prove that they exhibits blenders for some parametervalues. Using this family, we also prove the occurrence of blenders near certain non-transverse heterodimensional cycles under high regularity assumptions. The proof involves a renormalization scheme along heteroclinic orbits. We also investigate the connection between the blender and the original heterodimensional cycle.

Hénon-Like Families and Blender-Horseshoes at Nontransverse Heterodimensional Cycles

“…We consider the center-unstable Hénon-like family of endomorphisms G ξ,µ,κ1,κ2 : for some ε > 0. See [14] for a version of this result for blenders (instead of blenderhorseshoes) and [29] for a complete numerical analysis of this family including the study of the creation and annihilation of blenders.…”

“…The second type of dynamics was studied in [4] (to get infinitely many sinks/sources) and in [5] (to get universal dynamics), and the first type in [15] (where heterodimensional tangencies were introduced). In this paper, we consider heterodimensional tangencies and heterodimensional cycles associated to saddles with non-real eigenvalues, continuing the study started in [14], where a similar configuration (with saddles having real eigenvalues) was considered.…”

“…Our result state that, in our setting, the bifurcating diffeomorphisms f can be approximated by diffeomorphisms exhibiting blender-horseshoes, see Theorem 1 and Corollary 1. This approximation is done using a renormalisation scheme converging to a centerunstable Hénon-like family (see equations (2.1) and (2.2)) and using the fact that this family exhibits blender-horseshoes, see [16] and [14] for a preliminary version. In this approach, the Hénon-like family plays the role of the quadratic family in the homoclinic setting.…”

“…A part of this problem (concerning the generation of blenders) was considered and solved in [14], but the embedding of these blenders in the dynamics in [14] is still not completely adequate. The results in [14] have three parts, it is proved that: (i) There is a center-unstable Hénon-like family displaying (for appropriate range of parameters) blender-horseshoes (see also [16]); (ii) There is renormalisation scheme converging to that Hénon-like family; (iii) Some heteroclinic-like relations between the blender and the saddles in the cycle are obtained for appropriate C 1+α , α < 1, perturbations. Here we see how (i) and (ii) hold in our setting and in a forthcoming paper we deal with the problem of the dynamical embedding of the obtained blender and the heteroclinic relations, see also [41].…”

Robust Heteroclinic Tangencies

“…Indeed, the authors showed that the C 1 -unfolding of a three dimensional heterodimensional tangency (with k T = 1) leads to C 1robustly non-dominated dynamics and in some cases to very intermingled dynamics related to universal dynamics, for details see [9,6]. In the C r -topologies with r > 1, the bifurcation of such tangencies leads, for instance, to the existence of blender dynamics [8,10]. Kiriki and Soma in [12] obtain the first examples of C 2 -robust heterodimensional tangencies with c T = 1 and k T = d − 2 in any manifold of dimension d ≥ 3.…”