Entropy theorems along times when $x$ visits a set

Abstract

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 $x$.