Cesàro summability in a linear autonomous difference equation

Abstract

For a linear autonomous difference equation with a unique real eigenvalue  λ 0 \lambda _{0} , it is shown that for every solution  x x the ratio of x x and the eigensolution corresponding to  λ 0 \lambda _{0} is Cesàro summable to a limit which can be expressed in terms of the initial data. As a consequence, for most solutions the Lyapunov characteristic exponent is equal to  λ 0 \lambda _{0} . The proof is based on a Tauberian theorem for the Laplace transform.