Rochlin's Lemma

The lemma of Rokhlin 's mathematics is an important tool of ergodic theory .

In the theory of dynamic systems it is used to partition the phase space of a dynamic system and in this way to be able to study the dynamics of the system.

It was proven by Vladimir Rochlin in 1948 .

Independently of Rochlin, it was also proven by Shizuo Kakutani , which is why the term Kakutani-Rokhlin lemma is also used in the English-speaking world .

Statement of the lemma

Let be an atomless probability space and an invertible ergodic transformation . ${\displaystyle (X,\Omega ,\mu )}$ ${\displaystyle T\colon X\to X}$

Then there is a measurable subset for each and every one , so that ${\displaystyle \epsilon >0}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle A\subset X}$

${\displaystyle A,T(A),\ldots ,T^{n}(A)}$

are pairwise disjoint and

${\displaystyle \mu (\cup _{j=0}^{n}T^{j}(A))>1-\epsilon }$

applies, where is set. ${\displaystyle T^{0}=\mathrm {id} _{X}}$

literature

• V. Rokhlin: A "general" measure-preserving transformation is not mixing. (Russian) Doklady Akad. Nauk SSSR (NS) 60, (1948), pp. 349-351.
• Manfred Denker: Introduction to the Analysis of Dynamic Systems. Springer, Berlin / Heidelberg 2005, ISBN 3-540-20713-9 , p. 211.