Subsystems with the shadowing 性质 for bm$\mathbb{Z}^{k}$-actions

Abstract

In this paper, subsystems with shadowing 性质 for $\\mathbb{Z}^{k}$-actions are investigated. Let $\\alpha$ be a continuous $\\mathbb{Z}^{k}$-action on a compact metric space $X$. We introduce the notions of pseudo orbit and shadowing 性质 for $\\alpha$ along with subsets, particularly subspaces, of $\\mathbb{R}^{k}$. We show that if $\\alpha$ has the shadowing 性质 and is expansive along a subspace $V$ of $\\mathbb{R}^{k}$, then so does for $\\alpha$ along any subspace $W$ of $\\mathbb{R}^{k}$ containing $V$. Let $\\alpha$ be a smooth $\\mathbb{Z}^{k}$-action on a closed Riemannian manifold $M$, $\\mu$ an ergodic probability measure and $\\Gamma$ the Oseledec set. We show that, under a basic 假设 on the Lyapunov spectrum, $\\alpha$ has the shadowing 性质 and is expansive on $\\Gamma$ along any subspace $V$ of $\\mathbb{R}^{k}$ containing a regular vector; furthermore, $\\alpha$ has the quasi-shadowing 性质 on $\\Gamma$ along any 1-dimensional subspace $V$ of $\\mathbb{R}^{k}$ containing a first-type singular vector. As an application, we also study the 1-dimensional subsystems (i.e., flows) with shadowing 性质 for the $\\mathbb{R}^{k}$-action on the suspension manifold induced by $\\alpha$.