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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm222-3-4,
author = {Jonathan Meddaugh and Brian E. Raines},
title = {Shadowing and internal chain transitivity},
journal = {Fundamenta Mathematicae},
volume = {220},
year = {2013},
pages = {279-287},
zbl = {1294.37008},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm222-3-4}
}
Jonathan Meddaugh; Brian E. Raines. Shadowing and internal chain transitivity. Fundamenta Mathematicae, Tome 220 (2013) pp. 279-287. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm222-3-4/