Kac’s theorem

Let $f:M\to M$ 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 $E$:

$${\rho }_E(x)=\min \{n\ge 1:f^n(x)\in E\}$$

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

$$E_0=\{x\in E:f^n(x)\notin E,n\ge 1\}$$

$$E_0^{*}=\{x\in M:f^n(x)\notin E,n\ge 0\}$$

By Poincaré recurrence theorem, $\mu (E_0)=0$.Kac’s theorem asserts that the function ${\rho }_E$ is integrable and

$${\int }_E{\rho }_E𝑑\mu =\mu (M)-\mu (E_0^{*})$$

When the system is ergodic, then $\mu (E_0^{*})=0$, and Kac’s theorem implies:

$$\frac{1}{\mu (E)}{\int }_E{\rho }_E𝑑\mu =\frac{\mu (M)}{\mu (E)}$$

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