On the exponent of $G$-spaces and isovariant extensors

The equivariant version of the Curtis-Schori-West theorem is investigated. It is proved that for a nondegenerate Peano $G$-continuum $\mathbb X$ with an action of the compact abelian Lie group $G$, the exponent $\exp\mathbb X$ is equimorphic to the maximal equivariant Hilbert cube if and only if the free part $\mathbb X_{\mathrm{free}}$ is dense in $\mathbb X$. We also show that the latter is sufficient for the equimorphy of $\exp\mathbb X$ and $\mathbb Q$ in the case of an action of an arbitrary compact Lie group $G$. The key to the proof of these results lies in the theory of the universal $G$-space (in the sense of Palais).
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