Takens' Last Problem and strong pluripotency

TL;DR

This paper proves the existence of robust strongly pluripotent dynamical systems on entire hyperbolic invariant sets, specifically in the context of wild Smale horseshoes, and demonstrates their persistent properties.

Contribution

It introduces a combinatorial method to identify strongly pluripotent diffeomorphisms in a Newhouse domain and proves their $C^r$-robustness for the entire hyperbolic set.

Findings

Existence of a 2D diffeomorphism with a wild Smale horseshoe with $C^r$-robust strongly pluripotent properties.

All elements in a certain $C^r$ neighborhood are strongly pluripotent for the whole Smale horseshoe.

Properties like non-trivial physical measures or historic behavior are $C^r$-persistent in a dense subset of the neighborhood.

Abstract

We consider the concept of strong pluripotency of dynamical systems for a hyperbolic invariant set, as introduced in [KNS]. To the best of our knowledge, for the whole hyperbolic invariant set, the existence of robust strongly pluripotent dynamical systems has not been proven in previous studies. In fact, there is an example of strongly pluripotent dynamical systems in [CV01], but its robustness has not been proven. On the other hand, robust strongly pluripotent dynamical systems for some proper subsets of hyperbolic sets had been found in [KS17, KNS]. In this paper, we provide a combinatorial way to recognize strongly pluripotent diffeomorphisms in a Newhouse domain and prove that they are $C^r$-robust, $2\leq r< \infty$. More precisely, we prove that there is a 2-dimensional diffeomorphism with a wild Smale horseshoe which has a $C^r$ neighborhood $\mathcal{U}_0$ where all elements are strongly pluripotent for the whole Smale horseshoe. Moreover, it follows from the result that any property, such as having a non-trivial physical measure supported by the Smale horseshoe or having historic behavior, is $C^r$-persistent relative to a dense subset of $\mathcal{U}_0$.