## Illinois Journal of Mathematics

- Illinois J. Math.
- Volume 48, Number 1 (2004), 59-69.

### Entropy theorems along times when $x$ visits a set

Tomasz Downarowicz and Benjamin Weiss

#### 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 $x$.

#### Article information

**Source**

Illinois J. Math., Volume 48, Number 1 (2004), 59-69.

**Dates**

First available in Project Euclid: 13 November 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.ijm/1258136173

**Digital Object Identifier**

doi:10.1215/ijm/1258136173

**Mathematical Reviews number (MathSciNet)**

MR2048214

**Zentralblatt MATH identifier**

1035.37004

**Subjects**

Primary: 37A35: Entropy and other invariants, isomorphism, classification

Secondary: 28D05: Measure-preserving transformations 28D20: Entropy and other invariants 37B99: None of the above, but in this section

#### Citation

Downarowicz, Tomasz; Weiss, Benjamin. Entropy theorems along times when $x$ visits a set. Illinois J. Math. 48 (2004), no. 1, 59--69. doi:10.1215/ijm/1258136173. https://projecteuclid.org/euclid.ijm/1258136173