Fixed point properties of nilpotent group actions on 1-arcwise connected continua

Shi, Enhui; Sun, Binyong

doi:10.1090/s0002-9939-08-09522-1

Cited by 20 publications

(15 citation statements)

References 10 publications

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“…However this is not true even for actions on dendrites. Since solvable groups are amenable we have by a note in [22] that there is an amenable group action on a dendrite (even on an arc) without a fixed point. On the other hand, it was proved that in the case of countable amenable group actions on a dendrite [23], or even on a uniquely arcwise connected continuum [24] there is always a fixed point or a 2-periodic point.…”

Section: Resultsmentioning

confidence: 99%

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Every continuous action of a compact group on a uniquely arcwise connected continuum has a fixed point

Vejnar

2018

J. Fixed Point Theory Appl.

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We are dealing with the question whether every group or semigroup action (with some additional property) on a continuum (with some additional property) has a fixed point. One of such results was given in 2009 by Shi and Sun. They proved that every nilpotent group action on a uniquely arcwise connected continuum has a fixed point. We are seeking for this type of results with e.g. commutative, compact or torsion groups and semigroups acting on dendrites, dendroids, λ-dendroids and uniquely arcwise connected continua. We prove that every continuous action of a compact or torsion group on a uniquely arcwise connected continuum has a fixed point. We also prove that every continuous action of a compact and commutative semigroup on a uniquely arcwise connected continuum has a fixed point.

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Section: Resultsmentioning

confidence: 99%

“…Moreover we prove that a continuous action of a compact and commutative semigroup on a uniquely arcwise connected continuum has a fixed point too. This should be compared with the main result of [22] by which every nilpotent group action on a uniquely arcwise connected continuum has a fixed point.…”

Section: Introductionmentioning

confidence: 99%

Every continuous action of a compact group on a uniquely arcwise connected continuum has a fixed point

Vejnar

2018

J. Fixed Point Theory Appl.

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“…For groups actions on graphs and dendrites, several results related to minimality, sensitivity and existence of global fixed points have been obtained by some authors (see e.g., [23], [17], [24], [25]). For rigidity results for actions on dendrites, see [9].…”

Section: Introductionmentioning

confidence: 99%

Minimal sets and orbit spaces for group actions on local dendrites

Marzougui

Naghmouchi

2018

Math. Z.

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We consider a group G acting on a local dendrite X (in particular on a graph). We give a full characterization of minimal sets of G by showing that any minimal set M of G (whenever X is different from a dendrite) is either a finite orbit, or a Cantor set, or a circle. If X is a graph different from a circle, such a minimal M is a finite orbit. These results extend those of the authors for group actions on dendrites. On the other hand, we show that, for any group G acting on a local dendrite X different from a circle, the following properties are equivalent: (1) (G, X) is pointwise almost periodic.(2) The orbit closure relation R = {(x, y) ∈ X × X : y ∈ G(x)} is closed. (3) Every non-endpoint of X is periodic. In addition, if G is countable and X is a local dendrite, then (G, X) is pointwise periodic if and only if the orbit space X/G is Hausdorff.

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“…In fact, the topological dynamical systems of group actions on dendrites have been studied by some authors (see e.g. [4,8]). …”

Section: Introductions and Preliminariesmentioning

confidence: 99%

Minimal group actions on dendrites

Shi

Wang

Zhou

2010

Proc. Amer. Math. Soc.

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Fixed point properties of nilpotent group actions on 1-arcwise connected continua

Abstract: We show that every continuous action of a nilpotent group on a 1-arcwise connected continuum has at least one fixed point.

Cited by 20 publications

References 10 publications

Every continuous action of a compact group on a uniquely arcwise connected continuum has a fixed point

Every continuous action of a compact group on a uniquely arcwise connected continuum has a fixed point

Minimal sets and orbit spaces for group actions on local dendrites

Minimal group actions on dendrites

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