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Every continuous action of a compact group on a uniquely arcwise connected continuum has a fixed point. (English) Zbl 1403.54030
The main results. $$1)$$ Let $$G$$ be a group every element of which is contained in a compact subgroup. Then every continuous action of $$G$$ on a uniquely arcwise connected continuum has a fixed point. Moreover, the set of fixed points of the action is an arcwise connected continuum.
$$2)$$ Let $$G$$ be a compact commutative semigroup. Then every continuous action of $$G$$ on a uniquely arcwise connected continuum or on a tree-like continuum has a fixed point.

##### MSC:
 54H25 Fixed-point and coincidence theorems (topological aspects) 37B45 Continua theory in dynamics 54F15 Continua and generalizations 54H11 Topological groups (topological aspects)
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