Ergodic Theory - Sojourn Time

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 χ_A 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 R_i = k_i − k_i−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.