$C^1$-robust strong pluripotency for blender-horseshoes

Abstract

Suppose that $M$ is a closed manifold of dimension greater than two. We show that there exists a $C^1$-diffeomorphism $f_0:M\longrightarrow M$ with a wild affine blender-horseshoe $Λ_{f_0}$ such that any element $f$ of $\mathrm{Diff}^1(M)$ sufficiently $C^1$-close to $f_0$ is strongly pluripotent for the continuation $Λ_f^{(\mathrm{mj})}$ of $Λ_{f_0}^{(\mathrm{mj})}$, where $Λ_{f_0}^{(\mathrm{mj})}$ is the dense subset of $Λ_{f_0}$ consisting of elements with majority condition.