A class of spaces that admit no sensitive commutative group actions

Jiehua Mai¹ ; Enhui Shi²

¹ Institute of Mathematics Shantou University Shantou, Guangdong, 515063, P.R. China
² Department of Mathematics Soochow University Suzhou, Jiangsu, 215006, P.R. China

Fundamenta Mathematicae, Tome 217 (2012) no. 1, pp. 1-12

Résumé

We show that a metric space $X$ admits no sensitive commutative group action if it satisfies the following two conditions: (1) $X$ has property S, that is, for each $\varepsilon>0$ there exists a cover of $X$ which consists of finitely many connected sets with diameter less than $\varepsilon$; (2) $X$ contains a free $n$-network, that is, there exists a nonempty open set $W$ in $X$ having no isolated point and $n\in\mathbb N$ such that, for any nonempty open set $U\subset W$, there is a nonempty connected open set $V\subset U$ such that the boundary $\partial_X(V)$ contains at most $n$ points. As a corollary, we show that no Peano continuum containing a free dendrite admits a sensitive commutative group action. This generalizes some previous results in the literature.

Détail

DOI : 10.4064/fm217-1-1

Keywords: metric space admits sensitive commutative group action satisfies following conditions has property each varepsilon there exists cover which consists finitely many connected sets diameter varepsilon contains n network there exists nonempty set having isolated point mathbb nonempty set subset there nonempty connected set subset boundary partial contains points corollary peano continuum containing dendrite admits sensitive commutative group action generalizes previous results literature