Infinite dimensional hyperbolic dynamics

Abstract

Consider the infinite dimensional hyperbolic dynamical system given by the heat (semi-)flow on the loop space of a closed Riemannian manifold (in the presence of a generic perturbation). The recently discovered *backward* lambda-Lemma is expected to be the key substitute for the non-existing backward flow. The lack of a backward flow caused the field to dry out in the mid 80s after those results that only use a forward flow had been carried over from the finite dimensional situation. One main application will be to construct global stable foliations, because these -- together with the known unstable foliations -- turned out to be the fundamental tools in the finite dimensional case. In finite dimensions stable and unstable foliations were constructed by Palis in 1969. Their significance lies in the fact that they provide natural coordinate systems near hyperbolic singularities. Another major application of the backward lambda-Lemma will be to calculate the Morse homology associated to a *semi*-flow. This will generalize the method of Abbondandolo-Majer which only applies to genuine flows. In fact, our purpose is to make the Morse homology tool available to the field of geometric analysis where many examples of semi-flows arise. (AU)