Kac’s theorem

Let f:MM be a transformation and μ a finite invariant measure for f. Let E be a subset of M with positive measureMathworldPlanetmath. We define the first return map for E:


If the set on the right is empty, then we define ρE(x)=. The Poincaré recurrence theorem asserts that ρE is finite for almost every xR. We define the following sets:


By Poincaré recurrence theorem, μ(E0)=0. Kac’s theorem asserts that the function ρE is integrable and


When the system is ergodic, then μ(E0*)=0, and Kac’s theorem implies:


This equality can be interpreted as: the mean return time to E s inversely proportional to the measure of E.

Title Kac’s theorem
Canonical name KacsTheorem
Date of creation 2014-03-19 22:18:04
Last modified on 2014-03-19 22:18:04
Owner Filipe (28191)
Last modified by Filipe (28191)
Numerical id 4
Author Filipe (28191)
Entry type Theorem
Related topic Poincaré Recurrence theorem