2021/10/31 by Lin Wang, Xinsheng Wang, Wang, Lin +3
Mathematics · Physics and Astronomy · #37C50 #37C85 #37D30 #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.2111.00457
openalex publication_date 2021/10/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, subsystems with shadowing property for ℤk-actions are investigated. Let α be a continuous ℤk-action on a compact metric space X. We introduce the notions of pseudo orbit and shadowing property for α along subsets, particularly subspaces, of ℝk. Combining with another important property "expansiveness" for subsystems of α which was introduced and systematically investigated by Boyle and Lind, we show that if α has the shadowing property and is expansive along a subspace V of ℝk, then so does for α along any subspace W of ℝk containing V. Let α be a smooth ℤk-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 ℝk containing a regular vector; furthermore, α has the quasi-shadowing property on Γ along any 1-dimensional subspace V of ℝk containing a first-type singular vector. As an application, we also consider the 1-dimensional subsystems (i.e., flows) with shadowing property for the ℝk-action on the suspension manifold induced by α.