Asymptotic Behavior

In this chapter we show that all lower and upper Lyapunov exponents, both to the future and to the past, belong to some connected component of the tempered spectrum. We also consider the asymptotic behavior of a linear dynamics under exponentially decaying perturbations that can either be linear or nonlinear. It turns out that all Lyapunov exponents of the perturbed dynamics still belong to some connected component of the tempered spectrum of the linear dynamics. This result depends strongly on the use of Lyapunov norms, which are also used in other parts of the book.