Let \(f\) be a diffeomorphism of a smooth manifold \(M\). A nonempty compact \(f\)-invariant set \(K\subset M\) is called a quasi-attractor of \(f\) if:
(i) \(K\) admits a basis of open neighborhoods \(U\) such that \(f(\overline{U})\subset U\);
(ii) \(K\) is chain-transitive.
Denote by \(PH_{c=1}^1(M)\) the set of \(C^1\)-diffeomorphisms of \(M\) that preserve a partially hyperbolic decomposition \(TM=E^s\oplus E^c\oplus E^u\) with one-dimensional \(E^c\) and nonempty \(E^s\) and \(E^u\).
The authors prove that there exists an open and dense subset \(O\) of \(PH_{c=1}^1(M)\) such that any \(f\in O\) has finitely many quasi-attractors. This result is refined for three-dimensional manifolds.