  A classical problem in smooth dynamical systems is known as smooth
realization problem. It asks if given a compact manifold $M$, one can construct
a volume preserving diffeomorphism with prescribed ergodic properties. We study
the decay of correlations for certain dynamical systems with non-uniformly
hyperbolic attractors, which natural invariant measure is not the volume, but
the SRB measure. The system $g$ that we consider is produced by applying the
slow-down procedure to a uniformly hyperbolic diffeomorphism $f$ with an
attractor. Under certain assumptions on the map $f$ and the slow-down
neighborhood, we show that the map $g$ admits polynomial upper and lower bounds
on correlations with respect to its SRB measure and the class of H\"older
continuous potentials. Our results apply to the Smale-Williams solenoid, as
well as its sufficiently small perturbations.
