The Generalisations of Ergodic Theorems

Question

We all know that the first two ergodic theorems are Birkhoff ergodic theorem and Von-Neumann ergodic theorem. And we also have Wiener-Wintner ergodic theorem. Are these theorems all the ergodic theorems or there are other ergodic theorems ? What are the generalisations of classical ergodic theorems ? I would be grateful if one could answer my question.

Answer 1

There are a lot of different pointwise ergodic theorems. For example, one can prove that the sums $\dfrac{1}{N}\displaystyle\sum_{n\leq N-1}f(T^{n^2}x)$ converge as $N\rightarrow\infty$ for a.e. $x$, for $f∈L^2$, say.

I am working on this case currently and I have proved the first step $$\left|\left|\dfrac{1}{N}\displaystyle\sum_{n\leq N-1}f_n - \dfrac{1}{N}\displaystyle\sum_{n\leq N-1}f_{n+k}\right|\right|_{2} \leq \dfrac{2CK}{N},$$ for any $k = 0, \dots , K-1$, such that $(f_n)_{n\geq0}$ be a sequence of functions in $L^2$ and $||f_n||_2 \leq C$ for all $n$. How to prove the next step using Cauchy-Schwarz inequality: $$\left|\left|\dfrac{1}{N}\displaystyle\sum_{n\leq N-1}f_n\right|\right|_{2} \leq \left(\dfrac{1}{N}\displaystyle\sum_{n\leq N-1}\left|\left|\dfrac{1}{K}\displaystyle\sum_{k\leq K-1}f_{n+k}\right|\right|_{2}^{2}\right)^{\frac{1}{2}} + \dfrac{2CK}{N} \ ?$$