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PreprintarXiv licencearXiv

Topological Expansivity and Shadowing for Anosov Diffeomorphisms

  • Z. Li
  • C. A. Morales
  • S. Romaña
  • Y. Yang
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Abstract

We show that every Anosov diffeomorphism is topologically expansive in the sense of [9]. We also prove that Lipschitz shadowing forces the stable and unstable bundles to be uniformly transverse, with a quantitative bound in terms of the shadowing constant. Finally, we give two examples on complete Riemannian manifolds: one is expansive but does not have the shadowing property, while the other has finite volume, bounded sectional curvature, Lipschitz shadowing, and orthogonal invariant bundles, but is not expansive with respect to its Riemannian distance.

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Cite

BibTeXRIS
BibTeX
@misc{li2026topological,
  title = {{Topological Expansivity and Shadowing for Anosov Diffeomorphisms}},
  author = {Z. Li and C. A. Morales and S. Romaña and Y. Yang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.14276},
  eprint = {2609.14276},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.14276},
}
RIS
TY  - UNPB
TI  - Topological Expansivity and Shadowing for Anosov Diffeomorphisms
AU  - Z. Li
AU  - C. A. Morales
AU  - S. Romaña
AU  - Y. Yang
PY  - 2026
DA  - 2026/09/13
PB  - arXiv
DO  - 10.48550/arxiv.2609.14276
UR  - https://arxiv.org/abs/2609.14276
AB  - We show that every Anosov diffeomorphism is topologically expansive in the sense of [9]. We also prove that Lipschitz shadowing forces the stable and unstable bundles to be uniformly transverse, with a quantitative bound in terms of the shadowing constant. Finally, we give two examples on complete Riemannian manifolds: one is expansive but does not have the shadowing property, while the other has finite volume, bounded sectional curvature, Lipschitz shadowing, and orthogonal invariant bundles, but is not expansive with respect to its Riemannian distance.
ER  -