Uncountable $\omega $-limit sets with isolated points
Tom 205 / 2009
Fundamenta Mathematicae 205 (2009), 179-189
MSC: 37B45, 37E05, 54F15, 54H20.
DOI: 10.4064/fm205-2-6
Streszczenie
We give two examples of tent maps with uncountable (as it happens, post-critical) $\omega $-limit sets, which have isolated points, with interesting structures. Such $\omega $-limit sets must be of the form $C\cup R$, 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 $\omega $-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 $\omega $-limit set is uncountable. Secondly, we give an example of an $\omega $-limit set of the form $C\cup R$ for which the Cantor set $C$ is minimal.
