Entropy theorems along times when \(x\) visits a set (Q1425767)

Let \((T,X,\mu)\) be an ergodic measure-preserving transformation of a probability space and let \({\mathcal A}\) be a finite measurable partition of \(X\). Well-known ergodic theorems describe the limit behavior of ergodic averages of a test-function \(\phi\) along a trajectory of the map \(T\), i.e. \(\frac1n\sum_{k=0}^{n-1}\phi(T^nx)\). The paper deals with the following problem: fix a set \(B\) of positive measure and consider ergodic averages only at moments of time when the trajectory \(\{T^nx\}\) visits the set \(B\). In a version of the Shannon-McMillan-Breiman theorem the authors estimate the rate of the exponential decay of the measure of the cell containing the point \(x\) (for almost every \(x\)) and the exponential growth of the first return time of \(x\) to this cell of the partition obtained by observing the process only when it visits the set \(B\). It is shown also that a partition \({\mathcal A}\) with zero measure boundaries can be modified to an open cover so that the Shannon-McMillan-Breiman theorem still holds (up to a small error term) for this cover.