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:
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—Abraham Lincoln (1809–1865)
“And now in one hour’s time I’ll be out there again. I’ll raise my eyes and look down that corridor four feet wide with ten lonely seconds to justify my whole existence.”
—Colin Welland (b. 1934)