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PreprintCC BY 4.0arXiv

Hypercyclicity and Lipschitz-free operators

  • Christian Cobollo
  • Romuald Ernst
  • Quentin Menet
  • Alfred Peris
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Abstract

We study linearizations of dynamical systems and some of its topological properties. Special attention is paid to the case of Lipschitz-free operators, and they are shown to model the dynamics of very general linearizations. We provide a new criterion, called the targeting property, for the (weakly) mixing property in a linear dynamical system through a (possibly) non-linear restriction of it. We then apply this criterion to the linearization Tf of a Lipschitz map f: M→ M, where M is a metric space and the operator Tf is defined on the corresponding Lipschitz-free space ℱ (M)---the so-called Lipschitz-free operators. We show that, under some natural assumptions on the distance considered in M, the operator Tf is hypercyclic if and only if the map f has the targeting property.

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Cite

BibTeXRIS
BibTeX
@misc{cobollo2026hypercyclicity,
  title = {{Hypercyclicity and Lipschitz-free operators}},
  author = {Christian Cobollo and Romuald Ernst and Quentin Menet and Alfred Peris},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.17874},
  eprint = {2609.17874},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.17874},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Hypercyclicity and Lipschitz-free operators
AU  - Christian Cobollo
AU  - Romuald Ernst
AU  - Quentin Menet
AU  - Alfred Peris
PY  - 2026
DA  - 2026/09/15
PB  - arXiv
DO  - 10.48550/arxiv.2609.17874
UR  - https://arxiv.org/abs/2609.17874
AB  - We study linearizations of dynamical systems and some of its topological properties. Special attention is paid to the case of Lipschitz-free operators, and they are shown to model the dynamics of very general linearizations. We provide a new criterion, called the targeting property, for the (weakly) mixing property in a linear dynamical system through a (possibly) non-linear restriction of it. We then apply this criterion to the linearization T_f of a Lipschitz map f: M→ M, where M is a metric space and the operator T_f is defined on the corresponding Lipschitz-free space ℱ (M)---the so-called Lipschitz-free operators. We show that, under some natural assumptions on the distance considered in M, the operator T_f is hypercyclic if and only if the map f has the targeting property.
ER  -