Currents, flows and diffeomorphisms

Abstract

HERE we describe a geometrical object in a manifold which determines by integration of differential forms a closed current in the sense of de Rham. In certain examples from dynamical systems, for example Anosov diffeomorphisms, these geometrical objects will lead to non-trivial real homology classes. Our ‘geometrical currents’ share some of the features of incompressible flows in the manifold. A geometrical currentt is made up of three things-a partial foliation in the manifold which is oriented and provided with a transversal measure. To go into more detail, to give a partial foliation (the streamlines of the current) we give a closed subset S of our smooth manifold W divided into connected subsets L,. There is a collection of closed disks of the form