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Shadowing and internal chain transitivity

Tom 222 / 2013

Jonathan Meddaugh, Brian E. Raines Fundamenta Mathematicae 222 (2013), 279-287 MSC: 37B50, 37B10, 37B20, 54H20. DOI: 10.4064/fm222-3-4

Streszczenie

The main result of this paper is that a map $f:X\to X$ which has shadowing and for which the space of $\omega $-limits sets is closed in the Hausdorff topology has the property that a set $A\subseteq X$ is an $\omega $-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 $\omega $-limit set must also have the property that the space of $\omega $-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 $\omega $-limit set of a point, and we also show that topologically hyperbolic maps and certain quadratic Julia sets have a closed space of $\omega $-limit sets.

Autorzy

  • Jonathan MeddaughDepartment of Mathematics
    Baylor University
    Waco, TX 76798-7328, U.S.A.
    e-mail
  • Brian E. RainesDepartment of Mathematics
    Baylor University
    Waco, TX 76798-7328, U.S.A.
    e-mail

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