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Minimal actions of homeomorphism groups

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Gutman Y.

Fundamenta Mathematicae

2008

tom 198

nr 3

191-215

Let X be a closed manifold of dimension 2 or higher or the Hilbert cube. Following Uspenskij one can consider the action of Homeo(X) equipped with the compact-open topology on $Φ ⊂ 2^{2^{X}}$, the space of maximal chains in $2^{X}$, equipped with the Vietoris topology. We show that if one restricts the action to M ⊂ Φ, the space of maximal chains of continua, then the action is minimal but not transitive. Thus one shows that the action of Homeo(X) on $U_{Homeo(X)}$, the universal minimal space of Homeo(X), is not transitive (improving a result of Uspenskij). Additionally for X as above with dim(X) ≥ 3 we characterize all the minimal subspaces of V(M), the space of closed subsets of M, and show that M is the only minimal subspace of Φ. For dim(X) ≥ 3, we also show that (M,Homeo(X)) is strongly proximal.

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1 Fundamenta Mathematicae

1 Gutman Y.

1 2008

Wyniki wyszukiwania

Minimal actions of homeomorphism groups