Poincaré Recurrence Theorem
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Theorem
Let $\struct {X, \BB, \mu, T}$ be a measure-preserving dynamical system.
Then for each $A \in \BB$:
- $\ds \map \mu {A \setminus \bigcap_{N \mathop = 1}^\infty \bigcup_{n \mathop = N}^\infty T^{-n} \sqbrk A} = 0$
That is, for $\mu$-almost all $x\in A$ there are integers $0 < n_1 < n_2 < \cdots$ such that $\map {T^{n_i} }x \in A$.
Proof
Let:
- $\ds A_\infty := \bigcap_{N \mathop = 1}^\infty \bigcup_{n \mathop = N}^\infty T^{-n} \sqbrk A$
For $N \in \N$, let:
- $\ds A_N := \bigcup_{n \mathop = N}^\infty T^{-n} \sqbrk A$
so that:
- $\ds A_\infty = \bigcap _{N \mathop = 1} ^\infty A_N$
Now, we need to show:
- $\ds \map \mu {A \setminus A_\infty} = 0$
First, for all $N \in \N$:
\(\ds A_N\) | \(=\) | \(\ds \bigcup_{n \mathop = N}^\infty T^{-n} \sqbrk A\) | ||||||||||||
\(\ds \) | \(\supseteq\) | \(\ds \bigcup_{n \mathop = N+1}^\infty T^{-n} \sqbrk A\) | ||||||||||||
\(\ds \) | \(\supseteq\) | \(\ds A_{N + 1}\) |
On the other hand, for all $N \in \N$:
\(\ds T^{-1} \sqbrk {A_N}\) | \(=\) | \(\ds T^{-1} \sqbrk {\bigcup_{n \mathop = N}^\infty T^{-n} \sqbrk A}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \bigcup_{n \mathop = N}^\infty T^{-1} \sqbrk {T^{-n} \sqbrk A}\) | Preimage of Union under Mapping/Family of Sets | |||||||||||
\(\ds \) | \(=\) | \(\ds \bigcup_{n \mathop = N+1}^\infty T^{-n} \sqbrk A\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds A_{N + 1}\) |
Hence, because $T$ is a $\mu$-preserving transformation:
- $\map \mu {A_N} = \map \mu {T^{-1} \sqbrk {A_N} } = \map \mu {A_{N + 1} }$
Therefore:
\(\ds \map \mu {A \setminus A_\infty}\) | \(\le\) | \(\ds \map \mu {A_0 \setminus A_\infty}\) | Measure is Monotone, $A \subseteq A_0$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \mu {A_0 \setminus \bigcap_{N \mathop = 1}^\infty A_N}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \mu {\bigcup_{N \mathop = 1} ^\infty {A_0 \setminus A_N} }\) | De Morgan's Law | |||||||||||
\(\ds \) | \(\le\) | \(\ds \sum_{N \mathop = 1}^\infty \map \mu {A_0 \setminus A_N}\) | Measure is Countably Subadditive | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{N \mathop = 1}^\infty \paren {\map \mu {A_0} - \map \mu {A_N} }\) | Measure of Set Difference with Subset | |||||||||||
\(\ds \) | \(=\) | \(\ds 0\) |
$\blacksquare$
Source of Name
This entry was named for Jules Henri Poincaré.
Sources
- 2011: Manfred Einsiedler and Thomas Ward: Ergodic Theory: with a view towards Number Theory Theorem $2.11$
- 2013: Peter Walters: An Introduction to Ergodic Theory (4th ed.) Theorem $1.4$