Construction of differentiable flows with prescribed attractor

It was proved by Jack Segal and the author that a finite dimensional compactum can be an attractor of a flow on a topological manifold if and only if it has the shape of a finite polyhedron. There differentiability was not taken into account. It is the subject of the present paper to show that the same characterization applies to attractors of differentiable flows on manifolds of class Cr, 1 ⩽ r < ∞.