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Shadowing and internal chain transitivity

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The main result of this paper is that a map f: X → X which has shadowing and for which the space of ω-limits sets is closed in the Hausdorff topology has the property that a set A ⊆ X is an ω-limit set if and only if it is closed and internally chain transitive. Moreover, a map which has the property that every closed internally chain transitive set is an ω-limit set must also have the property that the space of ω-limit sets is closed. As consequences of this result, we show that interval maps with shadowing have the property that every internally chain transitive set is an ω-limit set of a point, and we also show that topologically hyperbolic maps and certain quadratic Julia sets have a closed space of ω-limit sets.
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  • Department of Mathematics, Baylor University, Waco, TX 76798-7328, U.S.A.
  • Department of Mathematics, Baylor University, Waco, TX 76798-7328, U.S.A.
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bwmeta1.element.bwnjournal-article-doi-10_4064-fm222-3-4
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