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Is there any demonstration purely dynamic of the Birkhoff Ergodic Theorem, i.e, without the Maximal Ergodic Theorem??

I ask this question because I never understood the intuition behind The Maximal Ergodic Theorem.

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There is an elementary proof due to Katznelson and Weiss which I found instructive. I do not know if that proof counts as "purely dynamic". – Martin Dec 30 '12 at 0:48
I have deleted some unconstructive comments. @HarryPotter: I would advise you to stay on-topic as you are learning your way around this site. "Joke" posts regarding your pseudonym (like the one I just deleted on a different question) will be considered spam. – Zev Chonoles Dec 30 '12 at 1:24
@Martin Could you give the main ideas of Katznelson and Weiss' proof in an answer? – Davide Giraudo Feb 21 '13 at 9:25

1 Answer 1

I know a proof of aubadditive ergodic theorem, without using the Birkhoff ergodic theorem. Thus, the Birkhoff ergodic theorem can be regarded as a Corollary of subadditive ergodic theorem.

Given the system $(M,f,\mu)$, we call a sequence $\varphi_n: M \to \Bbb R$ subadditive if $\varphi_{m+n} \leq \varphi_m + \varphi_n \circ f^m$.

The statement of the subadditive ergodic theorem is as follows:

Let $\varphi_n : M \to \Bbb R$, $n\ge 1$ be a subadditive sequence of measurable functions such that $\varphi_1^{+}\in L^1(\mu)$. Then the sequence $(\varphi_n/n)_n$ converges for $\mu$-almost every point to a measurable function $\varphi:M \to [-\infty,+\infty)$. Moreover, $\varphi^{+}\in L^1(\mu)$ and

$$ \int \varphi\,d\mu = \lim_n \frac{1}{n} \int \varphi_n\,d\mu = \inf_{n}\frac{1}{n} \int \varphi_n \,d\mu\in [-\infty , + \infty). $$

One easily sees that the Birkhorf ergodic theorem follows from the above theorem, since the sequence $\varphi_n=\sum_{i=0}^{n-1}\varphi\circ f^i$ is a subadditive function.

The idea of the proof is to consider the number sequence $a_n=\int \varphi_nd\mu$ and thus, by subadditivity, define $$L=\lim_n\frac{a_n}{n}=\inf_n\frac{a_n}{n}$$ Then define $$\varphi_-(x)=\liminf \frac{a_n}{n}; \varphi_+(x)=\limsup \frac{a_n}{n}$$ Finally, one tries to show that $$\int\varphi_-(x)d\mu\geq L\geq \int\varphi_+(x)dmu$$ and finishes the proof.

The proof following the above idea is given by A.Avila and J.Bochi, Proof of the subadditive ergodic theorem.

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