Shadowing, internal chain transitivity and α-limit sets

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  • Baylor University


Let f:X→X be a continuous map on a compact metric space X and let α f, ω f and ICT f denote the set of α-limit sets, ω-limit sets and nonempty closed internally chain transitive sets respectively. We show that if the map f has shadowing then every element of ICT f can be approximated (to any prescribed accuracy) by both the α-limit set and the ω-limit set of a full-trajectory. Furthermore, if f is additionally expansive then every element of ICT f is equal to both the α-limit set and the ω-limit set of a full-trajectory. In particular this means that shadowing guarantees that α f‾=ω f‾=ICT f (where the closures are taken with respect to the Hausdorff topology on the space of compact sets), whilst the addition of expansivity entails α ff=ICT f. We progress by introducing novel variants of shadowing which we use to characterise both maps for which α f‾=ICT f and maps for which α f=ICT f.


Original languageEnglish
Article number124291
Number of pages19
JournalJournal of Mathematical Analysis and Applications
Issue number1
Early online date3 Jun 2020
Publication statusPublished - 1 Nov 2020


  • shadowing, α-limit set, ω-limit set, Internally chain transitive, Expansive, pseudo-orbit, Shadowing, Pseudo-orbit