Thick attractors of boundary preserving diffeomorphisms

A diffeomorphism is said to have a thick attractor provided that its attractor has positive but not full Lebesgue measure. A set in a functional space is quasiopen, if it may be obtained from an open set by removing a countable number of hypersurfaces. We prove that there exists a quasiopen set in the space of boundary preserving diffeomorphisms of a compact manifold with boundary, such that any map in this set has a thick attractor. The meaning of the word “attractor” should be specified. In the above claim an “attractor” is, roughly speaking, a “topologically mixing maximal attractor”. We also conjecture that the claim is true for the Milnor attractor of diffeomorphisms and prove the claim for Milnor attractors of mild skew products. We reduce the conjecture above to a general conjecture about Milnor attractors of partially hyperbolic diffeomorphisms.