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A class of spaces that admit no sensitive commutative group actions

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Mai J., Shi E.

Fundamenta Mathematicae

2012

tom 217

nr 1

1-12

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 $∂_X(V)$ 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.

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1 Fundamenta Mathematicae

1 Mai J.

1 Shi E.

1 2012

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A class of spaces that admit no sensitive commutative group actions