We give two examples of tent maps with uncountable (as it happens, post-critical) ω-limit sets, which have isolated points, with interesting structures. Such ω-limit sets must be of the form C ∪ R, where C is a Cantor set and R is a scattered set. Firstly, it is known that there is a restriction on the topological structure of countable ω-limit sets for finite-to-one maps satisfying at least some weak form of expansivity. We show that this restriction does not hold if the ω-limit set is uncountable. Secondly, we give an example of an ω-limit set of the form C ∪ R for which the Cantor set C is minimal.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm205-2-6,
author = {Chris Good and Brian E. Raines and Rolf Suabedissen},
title = {Uncountable $\omega$-limit sets with isolated points},
journal = {Fundamenta Mathematicae},
volume = {205},
year = {2009},
pages = {179-189},
zbl = {1180.37020},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm205-2-6}
}
Chris Good; Brian E. Raines; Rolf Suabedissen. Uncountable ω-limit sets with isolated points. Fundamenta Mathematicae, Tome 205 (2009) pp. 179-189. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm205-2-6/