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Dynamical decompositions of
$\unicode[STIX]{x1D6FD}X\setminus X$
Published online by Cambridge University Press: 22 July 2015
Abstract
Let $X$ be a locally compact,
$\unicode[STIX]{x1D70E}$-compact, non-compact Hausdorff space and let
$\unicode[STIX]{x1D6FD}X$ be the Stone-Čech compactification of
$X$. Let
$G$ be a countably infinite discrete group continuously acting on
$X$, and suppose that, for every
$g\in G$,
$\text{Fix}(g)=\{x\in X:gx=x\}$ is compact. The action of
$G$ on
$X$ induces the action on
$\unicode[STIX]{x1D6FD}X$, and so on
$X^{\ast }=\unicode[STIX]{x1D6FD}X\setminus X$. Let
${\mathcal{D}}$ denote the finest decomposition of
$X^{\ast }$ into closed invariant subsets such that the corresponding quotient space of
$X^{\ast }$ is Hausdorff. Such a decomposition can be defined for any action of
$G$ on a compact Hausdorff space. Applying it to every member of
${\mathcal{D}}$ gives us a decomposition
${\mathcal{D}}^{2}$ of
$X^{\ast }$, then
${\mathcal{D}}^{3}$, and so on. We show that (1)
${\mathcal{D}}^{\unicode[STIX]{x1D714}_{1}}$ is the coarsest decomposition of
$X^{\ast }$ into closed invariant topologically transitive subsets, (2) there is a dense subset of points
$p\in X^{\ast }$ such that
$\overline{Gp}\in {\mathcal{D}}^{2}\setminus {\mathcal{D}}$, in particular,
${\mathcal{D}}$ is non-trivial and
${\mathcal{D}}^{2}$ is finer than
${\mathcal{D}}$, and (3) for every ordinal
$\unicode[STIX]{x1D6FC}<\unicode[STIX]{x1D714}_{1}$, there is
$p\in X^{\ast }$ such that
$\overline{Gp}\in {\mathcal{D}}^{\unicode[STIX]{x1D6FC}+2}\setminus {\mathcal{D}}^{\unicode[STIX]{x1D6FC}+1}$, so all the decompositions
${\mathcal{D}}^{\unicode[STIX]{x1D6FC}}$,
$\unicode[STIX]{x1D6FC}\leq \unicode[STIX]{x1D714}_{1}$, are distinct.
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- Research Article
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- © Cambridge University Press, 2015