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Flexibility of topological entropy and factor persistence in the C0-Gromov Hausdorff topology

  • Richard Cubas
  • Jorge Crisostomo
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Abstract

We prove that every surjective topological dynamical system on a compact metric space can be approximated in the C0-Gromov--Hausdorff topology by homeomorphisms with any prescribed topological entropy in [0,+∞], even when simultaneous approximation of its sets of invariant and ergodic measures is required. We also establish a rigidity result under uniform geometric bounds in a fixed dimension: for a homeomorphism of a closed, connected Riemannian manifold, topological stability in the sense of Gromov--Hausdorff guarantees its persistence as a factor of sufficiently close perturbations within the same geometric class. This persistence prevents topological entropy from decreasing and ensures that every invariant or ergodic measure of the original system admits a lift of the same type to the perturbed system.

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Cite

BibTeXRIS
BibTeX
@misc{cubas2026flexibility,
  title = {{Flexibility of topological entropy and factor persistence in the $C^0$-Gromov Hausdorff topology}},
  author = {Richard Cubas and Jorge Crisostomo},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.31939},
  eprint = {2610.31939},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.31939},
}
RIS
TY  - UNPB
TI  - Flexibility of topological entropy and factor persistence in the C^0-Gromov Hausdorff topology
AU  - Richard Cubas
AU  - Jorge Crisostomo
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2610.31939
UR  - https://arxiv.org/abs/2610.31939
AB  - We prove that every surjective topological dynamical system on a compact metric space can be approximated in the C^0-Gromov--Hausdorff topology by homeomorphisms with any prescribed topological entropy in [0,+∞], even when simultaneous approximation of its sets of invariant and ergodic measures is required. We also establish a rigidity result under uniform geometric bounds in a fixed dimension: for a homeomorphism of a closed, connected Riemannian manifold, topological stability in the sense of Gromov--Hausdorff guarantees its persistence as a factor of sufficiently close perturbations within the same geometric class. This persistence prevents topological entropy from decreasing and ensures that every invariant or ergodic measure of the original system admits a lift of the same type to the perturbed system.
ER  -