Some roughness results concerning reducibility for linear difference equations (Q1113394)

Consider the difference equation (1) \(x(n+1)=A(n)x(n)\), where A(n) is an invertible \(K\times K\) matrix function for \(n\in {\mathbb{Z}}\). Denote by \(| x|\) the Euclidean norm of the vector x. Let X(n) be a fundamental matrix solution of (1) and \(n,m\in {\mathbb{Z}}\). We say that X(n) has an exponential dichotomy with projection P, if there exist a projection P \((P^ 2=P)\) and constants \(K\geq 1\), \(a>0\) such that \((2)\quad | X(n)PX^{-1}(m)| \leq Ke^{-a(n-m)},n\geq m,\) \(| X(n)(I-P)X^{-1}(m)| \leq Ke^{-a(m-n)},m\geq n.\) The main result of this paper is the following. Suppose that (1) has a fundamental matrix solution X(n) which satisfies (2). Let B(n) be a matrix such that \(A(n)+B(n)\) is invertible and sup\(\{\) \(| B(n)|\), \(n\in {\mathbb{Z}}\}=\delta\), where \(\delta\) is sufficiently small. Then the perturbed equation (3) \(y(n+1)=(A(n)+B(n))y(n)\) has a fundamental matrix solution which also has an exponential dichotomy with the same projection P. A similar result is also obtained for the difference equation (1) which has the property (H) in the sense of \textit{K. J. Palmer} [J. Differ. Equations 36, 374-390 (1980; Zbl 0407.34036)].