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PreprintCC BY 4.0arXiv

Shadowing and Topological Stability via Generalized Pseudo-Hyperbolicity in Autonomous and Nonautonomous Linear Dynamics

  • Davor Dragicevic
  • Mihaly Pituk
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Abstract

Although generalized hyperbolicity has proved to be a fundamental notion in linear dynamics, recent results show that it is too restrictive to characterize the shadowing property of invertible bounded linear operators on Banach spaces. This motivates the introduction of generalized pseudo-hyperbolicity, a weaker notion formulated through continuous homogeneous maps generating exponentially decaying Green families. We prove that, for arbitrary sequences of invertible bounded linear operators on Banach spaces, generalized pseudo-hyperbolicity is equivalent to both shadowing and topological stability. In the autonomous setting, this yields the equivalence between shadowing and topological stability, removing additional assumptions from previous results. We also prove robustness of shadowing and topological stability for sequences of invertible bounded linear operators.

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Cite

BibTeXRIS
BibTeX
@misc{dragicevic2026shadowing,
  title = {{Shadowing and Topological Stability via Generalized Pseudo-Hyperbolicity in Autonomous and Nonautonomous Linear Dynamics}},
  author = {Davor Dragicevic and Mihaly Pituk},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.08426},
  eprint = {2610.08426},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.08426},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Shadowing and Topological Stability via Generalized Pseudo-Hyperbolicity in Autonomous and Nonautonomous Linear Dynamics
AU  - Davor Dragicevic
AU  - Mihaly Pituk
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2610.08426
UR  - https://arxiv.org/abs/2610.08426
AB  - Although generalized hyperbolicity has proved to be a fundamental notion in linear dynamics, recent results show that it is too restrictive to characterize the shadowing property of invertible bounded linear operators on Banach spaces. This motivates the introduction of generalized pseudo-hyperbolicity, a weaker notion formulated through continuous homogeneous maps generating exponentially decaying Green families. We prove that, for arbitrary sequences of invertible bounded linear operators on Banach spaces, generalized pseudo-hyperbolicity is equivalent to both shadowing and topological stability. In the autonomous setting, this yields the equivalence between shadowing and topological stability, removing additional assumptions from previous results. We also prove robustness of shadowing and topological stability for sequences of invertible bounded linear operators.
ER  -