GENERALIZED THEOREM OF HARTMAN-GROBMAN ON MEASURE CHAINS

We will prove the Theorem of Hartman-Grobman in a very general form. I t states the topological equivalence of the flow of a nonlinear non-au tonomous differential or difference equation with critical component t o the flow of a partially linearized equation. The critical spectrum h as not necessarily to be contained in the imaginary axis or the unit c ircle respectively. Further on we will employ the so-called calculus o n measure chains within dynamical systems theory. Within this calculus the usual one dimensional time scales can be replaced by measure chai ns which are essentially closed subsets of R. The paper can be underst ood without knowledge of this calculus. So our main theorem will be va lid even for equations defined on very strange time scales such as seq uences of closed intervals. This is especially interesting for applica tions within the theory of differential-difference equations or within numerical analysis of qualitative phenomena of dynamical systems.