Shift equivalence implies topological equivalence for some Anosov endomorphisms

The mathematical models for hyperbolic attractors such as the Lorenz attractor are Anosov endomorphisms on branched manifolds. In his study of the structure of expanding attractors, R. F. Williams shows that the weak topological equivalence for expanding attractors can be reduced to the easily studied shift equivalence. A more general question is whether this reduction also holds for general hyperbolic attractors. This paper shows that such a reduction does not hold. In fact it is proved that shift equivalence will imply topological equivalence for some Anosov endomorphisms. As a result, the structure of true hyperbolic attractors must be determined by weak topological equivalence.