For a continuous map f on a compact metric space (X,d), a set D ⊂ X is internally chain transitive if for every x,y ∈ D and every δ > 0 there is a sequence of points ⟨x = x₀,x₁,...,xₙ = y⟩ such that for 0 ≤ i< n. In this paper, we prove that for tent maps with periodic critical point, every closed, internally chain transitive set is necessarily an ω-limit set. Furthermore, we show that there are at least countably many tent maps with non-recurrent critical point for which there is a closed, internally chain transitive set which is not an ω-limit set. Together, these results lead us to conjecture that for tent maps with shadowing, the ω-limit sets are precisely those sets having internal chain transitivity.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm217-1-4,
author = {Andrew D. Barwell and Gareth Davies and Chris Good},
title = {On the $\omega$-limit sets of tent maps},
journal = {Fundamenta Mathematicae},
volume = {219},
year = {2012},
pages = {35-54},
zbl = {1251.37015},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm217-1-4}
}
Andrew D. Barwell; Gareth Davies; Chris Good. On the ω-limit sets of tent maps. Fundamenta Mathematicae, Tome 219 (2012) pp. 35-54. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm217-1-4/