Pluripotency of wandering dynamics

Abstract

This paper isolates a perturbative mechanism, which we call \emph{pluripotency}, by which the symbolic and statistical behavior of prescribed orbits in a uniformly hyperbolic set can be realized, after an arbitrarily small perturbation, along the forward orbits of all points in a set of positive Lebesgue measure. In this sense, pluripotency provides a way of reprogramming dynamics from both statistical and geometric viewpoints: the empirical measures of all points in a positive-measure set can be made to asymptotically follow those of a prescribed orbit in the hyperbolic set. We first give an abstract criterion, formulated in terms of symbolic itinerary descriptions, which is equivalent to a strong form of pluripotency. We then prove that this mechanism occurs robustly in higher-dimensional non-hyperbolic dynamics. More precisely, for every $2\le r<\infty$ and $\dim M\ge 3$, there exists a $C^r$-open set of diffeomorphisms with wild blender-horseshoes such that every diffeomorphism in this open set is strongly pluripotent for a dense invariant subset of the blender-horseshoe. As applications, this yields dense classes of diffeomorphisms with non-trivial Dirac physical measures and with historic wandering domains inside the same open set, providing a new mechanism related to Takens&#39; last problem.