The principal loop-bundle and dynamical systems

The purpose of this Note is to announce the proof of the non-existence of e xpansive homeomorphisms on simply connected closed manifolds, provided one assumes locally connectedness of local stable and unstable sets. We also sh ow the non-existence of homoclinic points in the universal covering of Anos ov diffeomorphisms. These results are obtained by studying the lifted dynam ics to the principal loop-bundle, thus obtaining a link between dynamical s ystems and universal gauge theory. (C) 1999 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.