• Corpus ID: 240354465

Subsystems with shadowing property for $\mathbb{Z}^{k}$-actions

@inproceedings{Wang2021SubsystemsWS,
  title={Subsystems with shadowing property for \$\mathbb\{Z\}^\{k\}\$-actions},
  author={Lin Wang and Xinsheng Wang and Yujun Zhu},
  year={2021}
}

• Lin Wang, Xinsheng Wang, Yujun Zhu
• Published 31 October 2021
• Mathematics

In this paper, subsystems with shadowing property for Z-actions are investigated. Let α be a continuous Z-action on a compact metric space X. We introduce the notions of pseudo orbit and shadowing property for α along subsets, particularly subspaces, of R. Combining with another important property “expansiveness” for subsystems of α which was introduced and systematically investigated by Boyle and Lind in [4], we show that if α has the shadowing property and is expansive along a subspace V of R, then so does for α along any subspace W of R containing V . Let α be a smooth Z-action on a closed Riemannian manifold M , μ an ergodic probability measure and Γ the Oseledec set. We show that, under a basic assumption on the Lyapunov spectrum, α has the shadowing property and is expansive on Γ along any subspace V of R containing a regular vector; furthermore, α has the quasi-shadowing property on Γ along any 1-dimensional subspace V of R containing a first-type singular vector. As an application, we also consider the 1-dimensional subsystems (i.e., flows) with shadowing property for the R-action on the suspension manifold induced by α.