Attracting sets on surfaces.

Let f be a continuous endomorphism of a surface M, and A an attracting set such that the restriction f|A : A → A is a d : 1 covering map. We show that if f is a local homeomorphism, then f is also a d : 1 covering of the immediate basin of A. Moreover, the techniques provide a characterization of invariant d : 1 continua on surfaces. These results are no longer true on manifolds of dimensions at least three.