We consider an ergodic measure-preserving system
in which we fix a measurable
partition $\mathcal{A}$ and a set $B$ of nontrivial measure. In
a version of the
Shannon-McMillan-Breiman Theorem, for almost every $x$, we
estimate the
rate of the exponential decay of the measure of the cell
containing $x$ of the partition
obtained by observing the process only at the times $n$ when
$T^nx\in B$. Next, we estimate
the rate of the exponential growth of the first return time
of $x$ to this cell.
Then we apply these estimates to topological dynamics. We
prove that a partition with
zero measure boundaries can be modified to an open cover so
that the S-M-B theorem
still holds (up to $\epsilon$) for this cover, and we
derive the \en\ \fu\ on \im s from
the rate of the exponential growth of the first return time
to the $(n,\epsilon)$-ball
around $x$.