  We show that if $f$ is an annular homeomorphism admitting an attractor which
is an irreducible annular continua with two different rotation numbers, then
the entropy of $f$ is positive. Further, the entropy is shown to be associated
to a $C^0$-robust rotational horseshoe. On the other hand, we construct
examples of annular homeomorphisms with such attractors so that the rotation
interval is uniformly large but the entropy approaches zero as much as desired.
  The developed techniques allow us to obtain similar results in the context of
Birkhoff attractors.
