Classify all Anosov systems or attractors (which $\Omega_i$ can occur as attractors?)
Some classification results for attractors: Grines, V.; Zhuzhoma, E. On structurally stable diffeomorphisms with codimension one expanding attractors. Trans. Amer. Math. Soc. 357 (2005), no. 2, 617–667. and Brown, Aaron W.(1-TUFT)
Nonexpanding attractors: conjugacy to algebraic models and classification in 3-manifolds.(English summary) J. Mod. Dyn. 4 (2010), no. 3, 517–548.