On the Residual Continuity of Global Attractors

In this brief paper, we studied the residual continuity of global attractors Aλ in varying parameters λ∈Λ with Λ a bounded Borel set in Rd. We first reviewed the well-known residual continuity result of global attractors and then showed that this residual continuity is equivalent to the dense continuity. Then, we proved an analogue continuity result in measure sense that, under certain conditions, the set-valued map λ↦Aλ is almost (in the Lebesgue measure sense) uniformly continuous: for any small ε>0 there exists a closed subset Cε⊂Λ with Lebesgue measure m(Cε)>μ(Λ)−ε such that the set-valued map ε↦Aε is uniformly continuous on Cε. This, in return, indicates that the selected attractors {Aλ:λ∈Cε} can be equi-attracting