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- W2275155009 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 $gin G$ , $text{Fix}(g)={xin X:gx=x}$ is compact. The action of $G$ on $X$ induces the action on $unicode[STIX]{x1D6FD}X$ , and so on $X^{ast }=unicode[STIX]{x1D6FD}Xsetminus X$ . 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 $pin X^{ast }$ 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 $pin X^{ast }$ 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." @default.
- W2275155009 created "2016-06-24" @default.
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- W2275155009 date "2015-07-22" @default.
- W2275155009 modified "2023-10-18" @default.
- W2275155009 title "Dynamical decompositions of" @default.
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- W2275155009 doi "https://doi.org/10.1017/etds.2015.37" @default.
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