Kac's lemma

In ergodic theory, Kac's lemma, demonstrated by mathematician Mark Kac in 1947, is a lemma stating that in a finite measure space the orbit of almost all the points contained in a set ' of such space, whose measure is ', return to ' within an average time inversely proportional to '.
The lemma extends what is stated by Poincaré recurrence theorem, in which it is shown that the points return in infinite times.

Formal statement

Let ' be a finite measure space. Let ' be a measurable transformation preserving '. Let be any measurable set with positive measure.
Define the first-return time function by
if this set is nonempty, and otherwise, let it be ' if no iterate of is in '.
Then one version of Kac's lemma states that ' is integrable, with
This version is provable using elementary measure theory and real analysis. If the system ' is in fact ergodic, then the set ' has zero measure, so, dividing both sides by ', we indeed get that the mean return time is equal to the measure of the whole space divided by the measure of the set ', which is the statement of the lemma.

Application

In physics, a dynamical system evolving in time may be described in a phase space, that is by the evolution in time of some variables. If this variables are bounded, that is having a minimum and a maximum, for a theorem due to Liouville, a measure can be defined in the space, having a measure space where the lemma applies. As a consequence, given a configuration of the system the average return period close to this configuration is inversely proportional to the considered size of volume surrounding the configuration.
Normalizing the measure space to 1, it becomes a probability space and the measure ' of its set ' represents the probability of finding the system in the states represented by the points of that set. In this case the lemma implies that the smaller is the probability to be in a certain state, the longer is the time of return near that state.
In formulas, if ' is the region close to the starting point and is the return period, its average value is:
Where is a characteristic time of the system in question.
Note that since the volume of ', therefore, depends exponentially on the variables in the system, decreases very rapidly as the variables of the system increase and consequently the return period increases exponentially.
In practice, as the variables needed to describe the system increase, the return period increases rapidly.