Entropy theorems along times when x visits a set

We consider an ergodic measure-preserving system in which we fix a measurable partition 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 Tⁿx ∈ 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 ε) for this cover, and we derive the entropy function on invariant measures from the rate of the exponential growth of the first return time to the (n, ε)-ball around x.