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

Abstract

In this paper, subsystems with shadowing property 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 property for $\alpha$ along subsets, particularly subspaces, of ${\mathbb{R}}^k$. Combining with another important property "expansiveness" for subsystems of $\alpha$ which was introduced and systematically investigated by Boyle and Lind, we show that if $\alpha$ has the shadowing property 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 assumption on the Lyapunov spectrum, $\alpha$ has the shadowing property and is expansive on $\Gamma$ along any subspace $V$ of ${\mathbb{R}}^k$ containing a regular vector; furthermore, $\alpha$ has the quasi-shadowing property on $\Gamma$ along any 1-dimensional subspace $V$ of ${\mathbb{R}}^k$ containing a first-type singular vector. As an application, we also consider the 1-dimensional subsystems (i.e., flows) with shadowing property for the ${\mathbb{R}}^k$-action on the suspension manifold induced by $\alpha$.

25 pages, Accepted by SCIENTIA SINICA Mathematica (in Chinese)

References in corpus (1)

• Quasi-Shadowing and Quasi-Stability for Dynamically Coherent Partially Hyperbolic Diffeomorphisms