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Regular homeomorphisms of $$\mathbb{R}^3$$ and of $$\mathbb{S}^3$$. (English) Zbl 1397.37025
The author shows that every compact abelian group of homeomorphisms of $$\mathbb{R}^3$$ is either zero-dimensional or equivalent to a subgroup of the orthogonal group O$$(3)$$. A similar result is proved for $$\mathbb{S}^3$$. Regular homeomorphisms that are conjugate to their inverses are also investigated.
##### MSC:
 37C85 Dynamics induced by group actions other than $$\mathbb{Z}$$ and $$\mathbb{R}$$, and $$\mathbb{C}$$ 37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.) 37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces 57S10 Compact groups of homeomorphisms
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