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PreprintarXiv licencearXiv

Weighted orbits of hypercyclic vectors on F-spaces

  • Jian Li
  • Yonghang Ruan
  • Yini Yang
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Abstract

We study ℱ-hypercyclicity of weighted orbits λnTnx for continuous linear operators T on complex F-spaces, where ℱ is a Furstenberg family with the cut--shift--and--paste (CuSP) property. We characterize the scalar sequences for which weighted hypercyclicity always implies ordinary hypercyclicity: their nonzero values must be bounded and bounded away from zero. The same condition removes weights for every such family. For the converse inclusion, we prove that a bounded scalar sequence preserves all ℱ-hypercyclic vectors if its backward-shift ω-limit system is distal and does not contain the zero sequence. For chaotic operators, minimality of this system suffices. In particular, every complex nilsequence which is not identically zero preserves ℱ-hypercyclicity, and equality holds when its nonzero values are bounded away from zero. We also obtain simultaneous weighted return results and stability under unimodular weights with asymptotically trivial multiplicative differences.

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Cite

BibTeXRIS
BibTeX
@misc{li2026weighted,
  title = {{Weighted orbits of hypercyclic vectors on F-spaces}},
  author = {Jian Li and Yonghang Ruan and Yini Yang},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.04629},
  eprint = {2611.04629},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.04629},
}
RIS
TY  - UNPB
TI  - Weighted orbits of hypercyclic vectors on F-spaces
AU  - Jian Li
AU  - Yonghang Ruan
AU  - Yini Yang
PY  - 2026
DA  - 2026/11/03
PB  - arXiv
DO  - 10.48550/arxiv.2611.04629
UR  - https://arxiv.org/abs/2611.04629
AB  - We study ℱ-hypercyclicity of weighted orbits λ_nT^nx for continuous linear operators T on complex F-spaces, where ℱ is a Furstenberg family with the cut--shift--and--paste (CuSP) property. We characterize the scalar sequences for which weighted hypercyclicity always implies ordinary hypercyclicity: their nonzero values must be bounded and bounded away from zero. The same condition removes weights for every such family. For the converse inclusion, we prove that a bounded scalar sequence preserves all ℱ-hypercyclic vectors if its backward-shift ω-limit system is distal and does not contain the zero sequence. For chaotic operators, minimality of this system suffices. In particular, every complex nilsequence which is not identically zero preserves ℱ-hypercyclicity, and equality holds when its nonzero values are bounded away from zero. We also obtain simultaneous weighted return results and stability under unimodular weights with asymptotically trivial multiplicative differences.
ER  -