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The Brin-Katok local entropy formula for non-compact random dynamical systems and an application

  • Zeng Lian
  • Jianhua Zhang
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Abstract

In this paper, we establish the classical Brin-Katok local entropy formula for C1 random dynamical systems on a complete Riemannian manifold (possibly non-compact) under a mild integrability condition. The proof depends on the constructions of measurable partitions adapted to asymptotically varying radii and finite measurable partitions with null-boundaries. As an application, under additional integrability assumptions, we obtain the Margulis-Ruelle inequality for C2 random dynamical systems on complete non-compact Riemannian manifolds. Furthermore, we also construct a counterexample showing that the integrability condition above cannot in general be omitted.

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Cite

BibTeXRIS
BibTeX
@misc{lian2026brinkatok,
  title = {{The Brin-Katok local entropy formula for non-compact random dynamical systems and an application}},
  author = {Zeng Lian and Jianhua Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.01896},
  eprint = {2611.01896},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.01896},
}
RIS
TY  - UNPB
TI  - The Brin-Katok local entropy formula for non-compact random dynamical systems and an application
AU  - Zeng Lian
AU  - Jianhua Zhang
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.01896
UR  - https://arxiv.org/abs/2611.01896
AB  - In this paper, we establish the classical Brin-Katok local entropy formula for C^1 random dynamical systems on a complete Riemannian manifold (possibly non-compact) under a mild integrability condition. The proof depends on the constructions of measurable partitions adapted to asymptotically varying radii and finite measurable partitions with null-boundaries. As an application, under additional integrability assumptions, we obtain the Margulis-Ruelle inequality for C^2 random dynamical systems on complete non-compact Riemannian manifolds. Furthermore, we also construct a counterexample showing that the integrability condition above cannot in general be omitted.
ER  -