  The question we deal with here, which was presented to us by Joe Auslander
and Anima Nagar, is whether there is a nontrivial cascade (X,T) whose
enveloping semigroup, as a dynamical system, is topologically weakly mixing
(WM). After an introductory section recalling some definitions and classic
results, we establish some necessary conditions for this to happen, and in the
final section we show, using Ratner's theory, that the enveloping semigroup of
the `time one map' of a classical horocycle flow is weakly mixing.
