$\mathbb{Z}^{k}$-作用一维子系统的Lipschitz跟踪性

Lipschitz Shadowing Property for 1-Dimensional Subsystems of $\mathbb{Z}^{k}$-Actions

中文关键词:  $\mathbb{Z}^{k}$-作用  Lipschitz跟踪性  一维子系统

英文关键词:$\mathbb{Z}^{k}$-actions  Lipschitz shadowing  1-dimensional subsystem

中文摘要:

      本文主要研究了$\mathbb{Z}^{k}$-作用一维子系统的跟踪性质. 文中运用两种等价的方式引入了$\mathbb{Z}^{k}$-作用一维子系统的伪轨以及跟踪性的概念. 对于一个闭黎曼流形上的光滑$\mathbb{Z}^{k}$-作用$T$, 我们通过诱导的非自治动力系统提出了Anosov方向的概念. 借助Bowen几何的方法, 我们证明了$T$沿着任意Anosov方向具有Lipschitz跟踪性.

英文摘要:

      In this paper, the shadowing property for 1-dimensional subsystems of $\mathbb{Z}^{k}$-actions is investigated. The concepts of pseudo orbit and shadowing property for 1-dimensional subsystems of $\mathbb{Z}^{k}$-actions are introduced in two equivalent ways. For a smooth $\mathbb{Z}^k$-action $T$ on a closed Riemannian manifold, we propose a notion of Anosov direction via the induced nonautonomous dynamical system. Adapting Bowen's geometric method to our case, we show that $T$ has the Lipschitz shadowing property along any Anosov direction.