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 ε > 0 there exists a cover of X which consists of finitely many connected sets with diameter less than ε; (2) X contains a free n-network, that is, there exists a nonempty open set W in X having no isolated point and n ∈ ℕ such that, for any nonempty open set U ⊂ W, there is a nonempty connected open set V ⊂ U such that the boundary 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm217-1-1,
author = {Jiehua Mai and Enhui Shi},
title = {A class of spaces that admit no sensitive commutative group actions},
journal = {Fundamenta Mathematicae},
volume = {219},
year = {2012},
pages = {1-12},
zbl = {1251.54032},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm217-1-1}
}
Jiehua Mai; Enhui Shi. A class of spaces that admit no sensitive commutative group actions. Fundamenta Mathematicae, Tome 219 (2012) pp. 1-12. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm217-1-1/