  This article reviews a generous sampling of both classical and more recent
results on the interplay between measurable and topological dynamics. In the
first part we have surveyed the strong analogies between ergodic theory and
topological dynamics as shown in the treatment of recurrence phenomena,
equicontinuity and weak mixing, distality and entropy. The prototypical result
of the second part is the statement that any abstract measure probability
preserving system can be represented as a continuous transformation of a
compact space, and thus in some sense ergodic theory embeds into topological
dynamics. The work also contains several new results. In particular (1) we
prove, for a Polish dynamical system, the equivalence of the existence of a
Borel cross-section with the coincidence of recurrence and periodicity; and (2)
for compact dynamical systems we provide a converse to the local variational
principle.
