  It is shown that the Ellis semigroup of a $\mathbb Z$-action on a compact
totally disconnected space is completely regular if and only if forward
proximality coincides with forward asymptoticity and backward proximality
coincides with backward asymptoticity. Furthermore, the Ellis semigroup of a
$\mathbb Z$- or $\mathbb R$-action for which forward proximality and backward
proximality are transitive relations is shown to have at most two left minimal
ideals. Finally, the notion of near simplicity of the Ellis semigroup is
introduced and related to the above.
