Birkhoff attractors arise from the study of dissipative annulus maps that twist the vertical direction. When their rotation set is nontrivial, these attractors exhibit a complicated topological structure, namely that of an indecomposable continuum. In this talk, we introduce a class of Birkhoff-like attractors for dissipative annulus maps and study the continuity properties of these attractors and their rotation sets under perturbations of the dynamics.