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 contin...

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Bibliographic Details
Published inMathematics (Basel) Vol. 10; no. 9; p. 1444
Main Authors Wang, Xingxing, Cui, Hongyong
Format Journal Article
LanguageEnglish
Published Basel MDPI AG 01.05.2022
Subjects
Algebra
Baire’s category
Continuity
Dynamical systems
Euclidean space
global attractor
residual continuity
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Summary: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.
ISSN:2227-7390
2227-7390
DOI:10.3390/math10091444