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