# Kac’s theorem

Let $f:M\rightarrow M$ be a transformation and $\mu$ a finite invariant measure for $f$. Let $E$ be a subset of $M$ with positive measure. We define the first return map for $E$:

 $\rho_{E}(x)=\min\{n\geq 1:f^{n}(x)\in E\}$

If the set on the right is empty, then we define $\rho_{E}(x)=\infty$. The PoincarÃ© recurrence theorem asserts that $\rho_{E}$ is finite for almost every $x\in R$. We define the following sets:

 $E_{0}=\{x\in E:f^{n}(x)\notin E,n\geq 1\}$
 $E_{0}^{*}=\{x\in M:f^{n}(x)\notin E,n\geq 0\}$

By PoincarÃ© recurrence theorem, $\mu(E_{0})=0$. Kac’s theorem asserts that the function $\rho_{E}$ is integrable and

 $\int_{E}\rho_{E}d\mu=\mu(M)-\mu(E_{0}^{*})$

When the system is ergodic, then $\mu(E_{0}^{*})=0$, and Kac’s theorem implies:

 $\frac{1}{\mu(E)}\int_{E}\rho_{E}d\mu=\frac{\mu(M)}{\mu(E)}$

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

Title Kac’s theorem KacsTheorem 2014-03-19 22:18:04 2014-03-19 22:18:04 Filipe (28191) Filipe (28191) 4 Filipe (28191) Theorem PoincarÃ© Recurrence theorem