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A class of spaces that admit no sensitive commutative group actions

100%
Mai J., Shi E.
Fundamenta Mathematicae
|
2012
|
tom 217
|
nr 1
1-12
EN
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.
2
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Topological conjugation classes of tightly transitive subgroups of Homeo₊(ℝ)

100%
Shi E., Zhou L.
Colloquium Mathematicum
|
2016
|
tom 145
|
nr 1
111-120
EN
Let ℝ be the real line and let Homeo₊(ℝ) be the orientation preserving homeomorphism group of ℝ. Then a subgroup G of Homeo₊(ℝ) is called tightly transitive if there is some point x ∈ X such that the orbit Gx is dense in X and no subgroups H of G with |G:H| = ∞ have this property. In this paper, for each integer n > 1, we determine all the topological conjugation classes of tightly transitive subgroups G of Homeo₊(ℝ) which are isomorphic to ℤⁿ and have countably many nontransitive points.
3
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The ping-pong game, geometric entropy and expansiveness for group actions on Peano continua having free dendrites

100%
Shi E., Wang S.
Fundamenta Mathematicae
|
2009
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tom 203
|
nr 1
21-37
EN
It is shown that each expansive group action on a Peano continuum having a free dendrite must have a ping-pong game, and has positive geometric entropy when the acting group is finitely generated. As a corollary, it is shown that each Peano continuum having a free dendrite admits no expansive nilpotent group actions.
4
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Topological transitivity of solvable group actions on the line ℝ

81%
Wang S., Shi E., Zhou L., Cairns G.
Colloquium Mathematicum
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2009
|
tom 116
|
nr 2
203-215
EN
Let ϕ:G → Homeo₊(ℝ) be an orientation preserving action of a discrete solvable group G on ℝ. In this paper, the topological transitivity of ϕ is investigated. In particular, the relations between the dynamical complexity of G and the algebraic structure of G are considered.
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