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# 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 Poincaré recurrence theorem asserts that $\rho_{E}$ is finite for almost every $x\in R$. We define the following sets:

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

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

By Poincaré 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 $E$.

Related:

Poincaré Recurrence theorem

Type of Math Object:

Theorem

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