TY - JOUR
T1 - Entropy theorems along times when x visits a set
AU - Downarowicz, Tomasz
AU - Weiss, Benjamin
PY - 2004
Y1 - 2004
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=3042773737&partnerID=8YFLogxK
U2 - 10.1215/ijm/1258136173
DO - 10.1215/ijm/1258136173
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AN - SCOPUS:3042773737
SN - 0019-2082
VL - 48
SP - 59
EP - 69
JO - Illinois Journal of Mathematics
JF - Illinois Journal of Mathematics
IS - 1
ER -