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PreprintarXiv licencearXiv

Local stable and unstable sets for random dynamical systems

  • Xue Liu
  • Xiao Ma
  • Xiaomin Zhou
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Abstract

We study local stable and unstable sets for two-sided continuous bundle random dynamical systems. For an ergodic invariant measure with positive fiber measure-theoretic entropy and positive fiberwise maximal Lyapunov exponent, we show that the fiber entropy is determined by the action of the random maps on local unstable sets, and establish a lower bound for the Hausdorff dimension of local unstable sets in terms of the ratio of entropy to the maximal fiber Lyapunov exponent. If the upper box dimension of the phase space is finite, we obtain a weak form of Ruelle's inequality.

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Cite

BibTeXRIS
BibTeX
@misc{liu2026local,
  title = {{Local stable and unstable sets for random dynamical systems}},
  author = {Xue Liu and Xiao Ma and Xiaomin Zhou},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2606.23381},
  eprint = {2606.23381},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.23381},
}
RIS
TY  - UNPB
TI  - Local stable and unstable sets for random dynamical systems
AU  - Xue Liu
AU  - Xiao Ma
AU  - Xiaomin Zhou
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2606.23381
UR  - https://arxiv.org/abs/2606.23381
AB  - We study local stable and unstable sets for two-sided continuous bundle random dynamical systems. For an ergodic invariant measure with positive fiber measure-theoretic entropy and positive fiberwise maximal Lyapunov exponent, we show that the fiber entropy is determined by the action of the random maps on local unstable sets, and establish a lower bound for the Hausdorff dimension of local unstable sets in terms of the ratio of entropy to the maximal fiber Lyapunov exponent. If the upper box dimension of the phase space is finite, we obtain a weak form of Ruelle's inequality.
ER  -