Z^(k)-作用的具有跟踪性的子系统

Subsystems with the shadowing property for Z^(k)-actions

本文研究Z^(k)-作用子系统的跟踪性.设α是紧致度量空间X上一个连续的Z^(k)-作用.我们引入α沿着R^(k)的子集(特别是R^(k)的子空间)的伪轨及跟踪性的概念,证明了如果α沿着R^(k)的子空间V具有跟踪性并且是可扩的,则α沿着R^(k)中任意包含V的子空间W也具有跟踪性并且是可扩的.设α是闭Riemann流形M上的一个光滑Z^(k)-作用,μ是一个遍历的概率测度,Γ是相应的Oseledec集.在关于Lyapunov谱的一个基本假设下,本文证明了α在Γ上沿着R^(k)中任意包含一个正则向量的子空间V具有跟踪性并且是可扩的.进一步得到α在Γ上沿着R^(k)中任意包含一个第一型奇异向量的1维子空间V具有拟跟踪性.作为应用,本文讨论了由α诱导的扭扩流形上的R^(k)-作用的具有跟踪性的1维子系统(即流).

In this paper,subsystems with the shadowing property for Z^(k)-actions are investigated.Letαbe a continuous Z^(k)-action on a compact metric space X.We introduce the notions of the pseudo orbit and the shadowing property forαalong with subsets,particularly subspaces,of R^(k).We show that ifαhas the shadowing property and is expansive along a subspace V of R^(k),then so does forαalong any subspace W of R^(k)containing V.Letαbe a smooth Z^(k)-action on a closed Riemannian manifold M,μbe an ergodic probability measure andΓbe the Oseledec set.We show that,under a basic assumption on the Lyapunov spectrum,αhas the shadowing property and is expansive onΓalong any subspace V of R^(k)containing a regular vector;furthermore,αhas the quasi-shadowing property onΓalong any 1-dimensional subspace V of R^(k)containing a first-type singular vector.As an application,we also study the 1-dimensional subsystems(i.e.,flows)with the shadowing property for the R^(k)-action on the suspension manifold induced by α.