中国科技论文在线 DISCRETE and CONTINUOUS Website: DYNAMICAL SYSTEMS Volume 12,Number2, February 2005 Pp
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http://www.paper.edu.cn 中国科技论文在线 DISCRETE AND CONTINUOUS Website: http://aimSciences.org DYNAMICAL SYSTEMS Volume 12,Number2, February 2005 pp. 355–362 SHADOWING IN RANDOM DYNAMICAL SYSTEMS He Lianfa, Zhu Yujun∗ College of Mathematics and Information Science Hebei Normal University, Shijiazhuang, 050016, P.R.China and Zheng Hongwen Institute of Mathematics and Physics North China Electric Power University, Beijing, 102206, P.R.China (Communicated by Lan Wen) Abstract. In this paper we consider the shadowing property for C1 random dynam- ical systems. We first define a type of hyperbolicity on the full measure invariant set which is given by Oseledec’s multiplicative ergodic theorem, and then prove that the system has the “Lipschitz” shadowing property on it. 1. Introduction. In the study of deterministic dynamical systems, shadowing property is not only an important dynamical property but also a powerful tech- nical tool. The research related to it involves many aspects of the modern theory of dynamical systems, such as the theory of stability (see [4], [8], [13], [15], [18], [19]), the existence of chaotic behavior (see [5], [6]), the numerical computation (see [17]), etc. One can see some other important results about shadowing property in [14] and [7]. As in the deterministic case, shadowing property also plays an important role in the study of random dynamical systems. For example, Gundlach and Kifer [2] use a shadowing theorem in the study of Markov partitions for random hyperbolic systems. In [16], Chow and Vleck give a shadowing lemma for random diffeomor- phisms (see Liu, Qian and Tang [10] for an alternative treatment). In this paper we shall consider the shadowing property for general C1 random dynamical systems. In section 2, we present the definition of random dynamical systems and the multi- plicative ergodic theorem. In section 3, we define a type of hyperbolicity on the full measure invariant set which is given by Oseledec’s multiplicative ergodic theorem. Section 4 is devoted to the proof of the main result. Everywhere in this paper, we assume that M is a smooth d-dimensional closed manifold (i.e., M is compact and without boundary). We denote by |·|, ·and d(·, ·) , respectively, the norm on TM, the operator norm and the metric on M induced by the Riemmanian metric. Since M is compact, we can take constant ρ0 > 0 such that for any x ∈ M, the exponential map expx : {v ∈ TxM : |v| < ∞ 2ρ0}−→M is a C diffeomorphism to the image. 2000 Mathematics Subject Classification. 37C50,37H99. Key words and phrases. random dynamical system, multiplicative ergodic theorem, shadowing property. *Corresponding author. Research Supported by the National Natural Science Foundation of China(No:10371030) and the Doctor’s Foundation of Hebei Normal University(No:L2003B05). 转载 355.