$C^1$-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 $f_0:M⟶M$ with a wild affine blender-horseshoe ${\Lambda }_{f_0}$ such that any element $f$ of ${\mathrm{Diff}}^1(M)$ sufficiently $C^1$-close to $f_0$ is strongly pluripotent for the continuation ${\Lambda }_f^{(\mathrm{mj})}$ of ${\Lambda }_{f_0}^{(\mathrm{mj})}$, where ${\Lambda }_{f_0}^{(\mathrm{mj})}$ is the dense subset of ${\Lambda }_{f_0}$ consisting of elements with majority condition.

26 pages