Extended hierarchies of invariant fiber bundles for dynamic equations on measure chains

Abstract

If a linear autonomous ordinary differential of difference equation possesses a coefficient operator, which is (pseudo-) hyperbolic or allows a more specific splitting of its spectrum into appropriate spectral sets, then this gives rise to a so-called hierarchy of invariant linear subspaces of X related to the ranges to the corresponding spectral projections. Together with the intersections of these invariant subspaces, we get an extended hierarchy. Each member of the hierarchy can be characterized dynamically as set of initial points for orbits with a certain asymptotic growth rate in forward or backward time. In this paper we show that such a scenario persists under perturbations w.r.t. two points of view: In the first instance, the invariant linear spaces become an “extended hierarchy” of invariant manifolds, if the linear part is perturbed by a globally Lipschitzian (or smooth) mapping on X. This will be done in the nonautonomous context of dynamic equations on measure chains or time scales, where the time-varying invariant manifolds are called invariant fiber bundles. Secondly, we derive perturbation results well-suited for up-coming applications in analytical discretization theory.