Pluripotency of wandering dynamics

TL;DR

This paper introduces the concept of pluripotency in dynamical systems, demonstrating its robustness and providing new insights into the behavior of non-hyperbolic diffeomorphisms with implications for Takens' last problem.

Contribution

It develops the theory of pluripotency, establishes practical conditions for it, and proves its $C^{r}$-robustness in non-hyperbolic diffeomorphisms, offering a new solution to Takens' last problem.

Findings

Pluripotency can be observed through small perturbations in dynamics.

The property is $C^{r}$-robust for certain non-hyperbolic systems.

Provides a new affirmative solution to Takens' last problem.

Abstract

This paper proposes a new concept of pluripotency inspired by Colli-Vargas [Ergod. Theory Dyn. Syst., 21(6):1657-1681, 2001] and presents fundamental theorems for developing the theory. Pluripotency reprograms dynamics from a statistical or geometrical point of view. This means that the dynamics of various codes, including non-trivial Dirac physical measures or historic behavior, can be observably and stochastically realized by arbitrarily small perturbations. We first give a practical condition equivalent to a stronger version of pluripotency. Next, we show that the property of pluripotency is $C^{r} (2\leq r<\infty)$-robust. Precisely, there exists a $C^{r}$-open set of non-hyperbolic diffeomorphisms that have wild blender-horseshoes and are strongly pluripotent. It implies a new affirmative solution to Takens' last problem for $C^{r}$ diffeomorphisms of dimension $n\geq 3$.