neighborhood U satisfying X = f~ f”U. We will study the real homology of hyperbolic n=0 expanding attractors (defined below in the appendix) using the branched manifolds of [S] and dynamical properties of [l] and [71. One can assume that X is connected and in an appropriate sense oriented (expanding attractors are locally homeomorphic to Euclidean space Cartesian product the Cantor set). We will replace f by a suitable power and then we have the