  In this paper we study generalized Poincar\'e-Andronov-Hopf bifurcations of
discrete dynamical systems. We prove a general result for attractors in
n-dimensional manifolds satisfying some suitable conditions. This result allows
us to obtain sharper Hopf bifurcation theorems for fixed points in the general
case and other attractors in low dimensional manifolds. Topological techniques
based on the notion of concentricity of manifolds play a substantial role in
the paper.
