FLOWS ON ORDERED BUNDLES

In the investigation of the stability of an equilibrium in a smooth strongly mono-tone dynamical system (semi ow), an indispensable tool is the Frobenius{Perron theorem (in the nite-dimensional case), and the Kren{Rutman theorem (in the in nite-dimensional case) applied to the linearization of the system at the equilib-rium. (For the theory of strongly monotone semi ows the reader is referred in the continuous-time case to e. g. [Hi], or [Sm{Th], in the discrete-time case to [Ta], and for the denition of C1 strongly monotone semi ows|to [Mi].) However, in many cases restricting oneself to equilibria does not suffice and one needs to consider the behavior of the linearization of the dynamical system on a more complex invariant set (for a recent analysis where it is necessary to consider any compact invariant set see [Po{Te]). To the author's knowledge, D. Ruelle was the rst (1979) to generalize the Frobenius{Perron theorem to the case (in our terminology) of strongly monotone linear discrete-time semi ows on a nite-dimensional strongly ordered vector bundle (see [R1], Proposition 3.2). The main result of the present paper (Theorem 1) is a generalization of Ruelle's result to a fairly broad class of strongly monotone semi ows on strongly ordered Banach bundles. The paper is organized as follows. Part 1 (Sections 1, 2 and 3) deals with abstract theorems about monotone linear semi ows on ordered Banach bundles