Uncountable $\omega $-limit sets with isolated points

Chris Good¹ ; Brian E. Raines² ; Rolf Suabedissen³

¹ School of Mathematics and Statistics University of Birmingham Birmingham, B15 2TT, UK
² Department of Mathematics Baylor University Waco, TX 76798-7328, U.S.A.
³ Mathematical Institute University of Oxford Oxford, OX1 3LB, UK

Fundamenta Mathematicae, Tome 205 (2009) no. 2, pp. 179-189.

Résumé

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.

Détail

DOI : 10.4064/fm205-2-6

Keywords: examples tent maps uncountable happens post critical omega limit sets which have isolated points interesting structures omega limit sets form cup where cantor set scattered set firstly known there restriction topological structure countable omega limit sets finite to one maps satisfying least weak form expansivity restriction does omega limit set uncountable secondly example omega limit set form cup which cantor set minimal