University of Georgia
Mathematics Department Colloquium
Professor Mate Wierdl
University of Tennessee, Memphis
November 14, 2002
Subsequence ergodic theorems
Abstract: Ergodic theory grew out of statistical mechanics, the statistical
description of matter. This latter means, for example, that instead of
describing the behavior of each individual water-molecule in a cup of
water, one is satisfied with finding the average speed, energy
etc. of the molecules. But then the fundamental question arises:
how can we measure the average speed or energy.
It is clearly impossible to measure the speed of each individual
molecule and then take the mean of the data. The ergodic theorem says
that it is enough to select a single molecule, measure its
speed in each second, and if we make enough measurements and take the
average of the data, the number will be basically the average speed of
all the molecules in the cup of water.
This amazing theorem has one drawback: it requires that the
measurements are taken exactly at every second. But in
practice, the measurements might be made at, say, 1, 3, 4, 6, 11, ...
seconds or, even worse, at 1.1, 2.4, 2.9, 4.3, ... seconds
instead of at 1, 2, 3 ... seconds. Obviously, we would like to know
whether we still can compute accurately the average speed from the
measured data.
This question might have been the original motivation to examine the
ergodic theorem along subsequences --- the topic of our talk. We
will see that the question also has connections to the theory of
uniform distribution mod 1.