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| Let $(X,T)$ be a flow, where $X$ is a compact Hausdorff space. Recall that by definition $(X,T)$ is distal if the proximal relation $P$ is trivial ($P=\Delta$) and an elementary argument shows that $(X,T)$ is equicontinuous if the regionally proximal relation $RP$ is trivial.
One would expect the following to be the case:
THEOREM.
Let $\pi:X \to Y$ be a homomorphism (continuous surjective equivariant map).
If $\pi(P)=\Delta$ then $(Y,T)$ is distal, and if $\pi(RP)=\Delta$ then $(Y,T)$ is equicontinuous.
An immediate consequence is that for any flow the distal structure relation is the closed invariant equivalence relation generated by
$P$, and similarly for the equicontinuous structure relation with $RP$.
The theorem is well known (and true) but I have not seen it in print. Moreover, surprisingly, the proof is not immediate. |
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