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 , the space of maximal chains in , 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 , 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm198-3-1,
author = {Yonatan Gutman},
title = {Minimal actions of homeomorphism groups},
journal = {Fundamenta Mathematicae},
volume = {201},
year = {2008},
pages = {191-215},
zbl = {1156.37002},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm198-3-1}
}
Yonatan Gutman. Minimal actions of homeomorphism groups. Fundamenta Mathematicae, Tome 201 (2008) pp. 191-215. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm198-3-1/