The Princeton Companion to Mathematics by

Language: eng
Format: azw3
ISBN: 9781400830398
Publisher: Princeton University Press
Published: 2010-07-17T16:00:00+00:00

1.2 The Mean Ergodic Theorem

A beautiful application of the spectral theorem was found by VON NEUMANN [VI.91]. Imagine a checkerboard on which are distributed a certain number of checkers. Imagine that for each square there is designated a “successor” square (in such a way that no two squares have the same successor), and that every minute the checkers are rearranged by moving each one to its successor square. Now focus attention on a single square and each minute record with a 1 or 0 whether or not there is a piece on the square. This produces a succession of readings R1, R2, R3, . . . like this:

00100110010110100100 · · ·.

We might expect that over time, the average number of positive readings RJ = 1 will converge to the number of pieces on the board divided by the number of squares. If the rearrangement rule is not complicated enough, then this will not happen. For example, in the most extreme case, if the rule designates each square

as its own successor, then the readout will be either 00000 · · · or 111111 · · ·, depending on whether or not we chose a square with a piece on it to begin with. But if the rule is sufficiently complicated, then the “time average” (1/n) will indeed converge to the number of pieces on the board divided by the number of squares, as expected.

The checkerboard example is elementary, since in fact the only “sufficiently complicated” rules in this finite case are cyclic permutations of the squares of the board, and thus all the squares move past our observation post in succession. However, there are related examples where one observes only a small fraction of the data. For instance, replace the set of squares on a checkerboard with the set of points on a circle, and in place of the checkers, imagine that a subset S of a circle is marked as occupied. Let the rearrangement rule be the rotation of points on the circle through some irrational number of degrees. Stationed at a point x of the circle, we record whether x belongs to S, the first rotated copy of S, the second rotated copy of S, and so on to obtain a sequence of 0 or 1 readings as before. One can show that (for nearly every x) the time average of our observations will converge to the proportion of the circle occupied by S.

Similar questions about the relationship between time and space averages had arisen in thermodynamics and elsewhere, and the expectation that time and space averages should agree when the rearrangement rule is sufficiently complex became known as the ergodic hypothesis.

Von Neumann brought operator theory to bear on this question in the following way. Let H be the Hilbert space of functions on the squares of the checkerboard, or the Hilbert space of square-integrable functions on the circle. The rearrangement rule gives rise to a unitary operator U on H by means of the formula

(Uf)(y) = f(ϕ-1(y)),

where ϕ is the function describing the rearrangement.

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.