Only rational homology spheres admit $\Omega(f)$ to be union of DE   attractors
  If there exists a diffeomorphism $f$ on a closed, orientable $n$-manifold $M$ such that the non-wandering set $\Omega(f)$ consists of finitely many orientable $(\pm)$ attractors derived from expanding maps, then $M$ must be a rational homology sphere; moreover all those attractors are of topological dimension $n-2$.   Expanding maps are expanding on (co)homologies. 
Fan Ding, Jianzhong Pan, Shicheng Wang and Jiangang Yao