Recurrence theorem from Kac

In ergodic theory , one of the sub-areas of mathematics , Kac's return theorem deals with the question after which mean return time in discrete ergodic systems of a probability space the elements of certain measurable sets return to these sets for the first time. This theorem goes back to a scientific work by the mathematician Marek Kac (1914–1984) from 1947 and follows on from Poincaré's return theorem .

Formulation of the sentence

The sentence can be summarized as follows:

A probability space and an ergodic transformation are given . ${\displaystyle (X,\Sigma ,P)}$ ${\displaystyle (X,\Sigma ,P)}$ ${\displaystyle T\colon X\to X}$

Furthermore, a measurable amount is given and it applies . ${\displaystyle A\in \Sigma }$ ${\displaystyle P(A)>0}$

Then the equation applies with regard to the mean return time

${\displaystyle \int _{A}n_{A}(x)\,\mathrm {d} P_{A}(x)={\frac {1}{P(A)}}}$.

Explanations and Notes

• For and considering the value as the return period, with the after first returns . The numerical function given in this way is a - almost everywhere finite and - integrable function . ${\displaystyle A\in \Sigma }$ ${\displaystyle x\in A}$ ${\displaystyle n_{A}(x):=\inf\{\,m\in \mathbb {N} \mid m\geq 1\,,\,T^{m}(x)\in A\,\}}$ ${\displaystyle x}$ ${\displaystyle A}$ ${\displaystyle n_{A}\colon A\to \mathbb {N} \cup \{\infty \}}$ ${\displaystyle P}$ ${\displaystyle P}$
• For is it in limited measure . ${\displaystyle A\in \Sigma }$ ${\displaystyle P_{A}}$ ${\displaystyle A}$
• In the English-language specialist literature, the above recurrence clause is referred to as Kac's recurrence theorem or sometimes simply as Kac's theorem .

literature

• Gilbert Helmberg, Fred H. Simons: A dualization of Kac's recurrence theorem . In: Indagationes Mathematicae . tape 28 , 1966, pp. 608-615 ( MR0224772 ).
• Konrad Jacobs (Ed.): Selecta Mathematica. IV (=  Heidelberg Pocket Books . Volume 98 ). Springer-Verlag , Berlin, Heidelberg, New York 1972, ISBN 3-540-05782-X .
• M. Kac: On the notion of recurrence in discrete stochastic processes . In: Bulletin of the American Mathematical Society . tape 53 , 1947, pp. 1002-1010 ( MR0022323 ).
• Mark Pollicott, Michiko Yuri: Dynamical Systems and Ergodic Theory . Transferred to digital printing 2008 (=  London Mathematical Society Student Texts . Volume 40 ). Cambridge University Press , Cambridge 1998, ISBN 978-0-521-57294-1 ( MR1627681 ).

Individual evidence

1. ^ Selecta Mathematica. IV (Ed. Konrad Jacobs) 1972, pp. 46-56
2. ^ Mark Pollicot, Michiko Yuri: Dynamical Systems and Ergodic Theory. 1998, pp. 91-97
3. ^ Selecta Mathematica. IV, p. 49
4. ↑ Pollicot / Yuri, op.cit., P. 92