Abstract

In this paper, we consider the Lyapunov spectrum of perturbed random difference systems of the form x(n-r l) : A(n)x(n) + ((n), x(0) : ;s. e Rd, n:0,1,2,..., where A : (A(n))n>o is an ergodic stationary matrix-valued sequence and ((n) is a random perturbation. It is well known by the famous multiplicative ergodic theorem due to Oseledec and Millionshchikov [3, 4, 5] that the Lyapunov spectrum of the unperturbed system corresponding to (l), i.e., x(n+l):A(n)x(n) consists of r real numbers )q<)z<... <)r,, r < d. A wide class of random noise ((n) will be characterized so that this perturbed system has the unique Lyapunov exponent ,t : max(0, 1,).