45 research outputs found
Topology of the isometry group of the Urysohn space
Using classical results of infinite-dimensional geometry, we show that the
isometry group of the Urysohn space, endowed with its usual Polish group
topology, is homeomorphic to the separable Hilbert space. The proof is basedon
a lemma about extensions of metric spaces by finite metric spaces, which wealso
use to investigate (answering a question of I. Goldbring) the relationship,
when A,B are finite subsets of the Urysohn space, between the group of
isometries fixing A pointwise, the group of isometries fixing B pointwise, and
the group of isometries fixing the intersection of A and B pointwise
Isometrisable group actions
Given a separable metrisable space X, and a group G of homeomorphisms of X,
we introduce a topological property of the action of G on X which is equivalent
to the existence of a G-invariant compatible metric on X. This extends a result
of Marjanovic obtained under the additional assumption that X is locally
compact
Dynamical simplices and minimal homeomorphisms
We give a characterization of sets K of probability measures on a Cantor
space X with the property that there exists a minimal homeomorphism g of X such
that the set of g-invariant probability measures on X coincides with K. This
extends theorems of Akin (corresponding to the case when K is a singleton) and
Dahl (when K is finite-dimensional). Our argument is elementary and different
from both Akin's and Dahl's
Extensions of generic measure-preserving actions
We show that, whenever Gamma is a countable abelian group and Delta is a
finitely-generated subgroup of Gamma, a generic measure-preserving action of
Delta on a standard atomless probability space (X,mu) extends to a free
measure-preserving action of Gamma on (X,mu). This extends a result of Ageev,
corresponding to the case when Delta is infinite cyclic