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PreprintCC BY 4.0arXiv

Pluripotency and Anosov Dynamics

  • Shin Kiriki
  • Xiaolong Li
  • Kodai Yamamoto
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Abstract

We study two notions of pluripotency for dynamical systems. Strong pluripotency refers to the possibility of reprogramming, by arbitrarily small perturbations, the forward orbits of points in a set of positive Riemannian volume so that they track a prescribed orbit in a uniform time-averaged sense. In contrast, pluripotency, also called ``statistical pluripotency" in this paper, requires only asymptotic agreement of empirical measures. Strong pluripotency implies statistical pluripotency, but the converse need not hold. For Cr transitive Anosov diffeomorphisms, r≥2, we prove that strong pluripotency fails for every nonempty reference set. On the other hand, statistical pluripotency holds for a nonempty reference set if and only if it is contained in the basin of the unique SRB measure. This basin is invariant and has full Riemannian volume, but contains no periodic points. These results reveal a sharp distinction between the two notions even within uniformly hyperbolic dynamics.

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Cite

BibTeXRIS
BibTeX
@misc{kiriki2026pluripotency,
  title = {{Pluripotency and Anosov Dynamics}},
  author = {Shin Kiriki and Xiaolong Li and Kodai Yamamoto},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.30636},
  eprint = {2610.30636},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.30636},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Pluripotency and Anosov Dynamics
AU  - Shin Kiriki
AU  - Xiaolong Li
AU  - Kodai Yamamoto
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2610.30636
UR  - https://arxiv.org/abs/2610.30636
AB  - We study two notions of pluripotency for dynamical systems. Strong pluripotency refers to the possibility of reprogramming, by arbitrarily small perturbations, the forward orbits of points in a set of positive Riemannian volume so that they track a prescribed orbit in a uniform time-averaged sense. In contrast, pluripotency, also called ``statistical pluripotency" in this paper, requires only asymptotic agreement of empirical measures. Strong pluripotency implies statistical pluripotency, but the converse need not hold. For C^r transitive Anosov diffeomorphisms, r≥2, we prove that strong pluripotency fails for every nonempty reference set. On the other hand, statistical pluripotency holds for a nonempty reference set if and only if it is contained in the basin of the unique SRB measure. This basin is invariant and has full Riemannian volume, but contains no periodic points. These results reveal a sharp distinction between the two notions even within uniformly hyperbolic dynamics.
ER  -