Birkhoff sum converges

I am master student.I have been starting to learn Ergodic theory.

Let $(X,A,\mu,T)$ be a measure-preserving dynamical system.

Birkhoff Ergodic theorem

$$\lim_{n\to\infty} \frac{1}{n}\sum_{i=0}^{n-1}g(T^{i}(x))$$ (is called Birkhoff average of $g$) for every $g\in L^{1}(\mu)$ converges a.e.(every where if $g$ is continuous)

I have the following question:

Do we have thereom like Birkhoff ergodic theorem such that it say Birkhoff sum converges every where?

• @DanRust ,Mean Ergodic theorem deals with the $L^{2}$limiting behaviour of Birkhoff avreges for $g \in L^{2}$,what does it relate to Birkhoff sum? – Adam Aug 7 at 16:25
• You asked for a theorem like Birkhoff. In order to beat Birkhoff (for instance being valid for all $x$) you're going to have to weaken some conditions, such as the class of allowed functions. The Mean ergodic theorem is one such way. – Dan Rust Aug 7 at 16:37
• Yes,That is true,But i asked a theorem like Birkhoff theorem for Birkhoff sum NOT Birkhoff averges.Do you know such a theorem? – Adam Aug 7 at 16:50

Assume that the system $(X,A,\mu,T)$ is ergodic. The a. e. convergence of the Birkhoff sums would require $\int g\, d\mu = 0$. But, even under this assumption we do not need to have convergence a.e.
Namely, Ulrich Krengel in his 1978 paper On the speed of convergence in the ergodic theorem proved that for any ergodic measure preserving invertible transformation $T$ on $[0,1]$ with Lebesgue measure $\lambda$ and any positive sequence $\alpha_n \to 0$ as $n \to \infty$ one can find (even a continuous) function $g$, with $\int g\, d\lambda = 0$, and such that $$\limsup_{n\to\infty} \frac{1}{\alpha_n} \biggl\lvert \frac{1}{n} \sum\limits_{i=1}^{n-1} g(T^i(x)) \biggr\rvert = \infty$$ almost everywhere. Taking $\alpha_n=1/n$ we have $$\limsup_{n\to\infty}\, \biggl\lvert \sum\limits_{i=1}^{n-1} g(T^i(x)) \biggr\rvert = \infty$$ almost everywhere.
• You can see an example in Krengel's paper (as the author mentioned, the main point is that $T$ is aperiodic). Perhaps there are simpler examples: I do not know. – user539887 Aug 8 at 18:26