Let (X, Σ, μ) be a measure space such that μ(X) is finite and nonzero. The time spent in a measurable set A is called the sojourn time. An immediate consequence of the ergodic theorem is that, in an ergodic system, the relative measure of A is equal to the mean sojourn time:
for all x except for a set of measure zero, where χA is the indicator function of A.
The occurrence times of a measurable set A is defined as the set k1, k2, k3, ..., of times k such that Tk(x) is in A, sorted in increasing order. The differences between consecutive occurrence times Ri = ki − ki−1 are called the recurrence times of A. Another consequence of the ergodic theorem is that the average recurrence time of A is inversely proportional to the measure of A, assuming that the initial point x is in A, so that k0 = 0.
(See almost surely.) That is, the smaller A is, the longer it takes to return to it.
Read more about this topic: Ergodic Theory
Famous quotes containing the words sojourn and/or time:
“I must sojourn once to the ballot-box before I die. I hear the ballot-box is a beautiful glass globe, so you can see all the votes as they go in. Now, the first time I vote I’ll see if the woman’s vote looks any different from the rest—if it makes any stir or commotion. If it don’t inside, it need not outside.”
—Sojourner Truth (c. 1797–1883)
“He that will not apply new remedies, must expect new evils: for Time is the greatest innovator: and if Time, of course, alter things to the worse, and wisdom and counsel shall not alter them to the better, what shall be the end?”
—Francis Bacon (1561–1626)