A map $f: X \to X$ preserves an ergodic probability $\mu$, i.e., $\mu \circ f^{-1}=\mu$ and for any $\phi: X \to \mathbb{R}$ with $\int \phi d\mu=0$,
$$\frac{1}{n} \sum\_{i \le n} \phi \circ f^i \to 0 \text{ almost surely and in } L^1(\mu).$$


Therefore, $\max\_{n \ge N} \frac{1}{n} \sum\_{i \le n} \phi \circ f^i \to 0$ almost surely.


Similar to maximal inequality, are there references to quantitatively study


$$\max\_{n \ge N} \frac{1}{n} \sum\_{i \le n} \phi \circ f^i \text{ ?}$$


Similar to martingale inequality, are there references to study


$$\lVert\max\_{n \ge N} \frac{1}{n} \sum\_{i \le n} \phi \circ f^i\rVert\_{L^1} \precsim \lVert\frac{1}{N} \sum\_{i \le N} \phi \circ f^i\rVert\_{L^1} \text{ ?}$$



Maximal ergodic inequality
For a nice introductory discussion to the maximal ergodic inequality, see
[1]. In particular, inequality (5) there is Wiener's maximal ergodic theorem. See also Lemma 15.3 in [2]. A more advanced account, that in particular includes the $L^p$ maximal inequalities, is in the book [3], see Cor 2.2 page 8 for the Wiener inequality and the proof of Theorem 6.3 page 52 for the $L^p$ inequality


$$\lVert\max\_{n \le N} \frac{1}{n} \sum\_{i \le n} \phi \circ f^i\rVert\_{L^p} \le \frac{p}{p-1} \lVert \phi \rVert\_{L^p} \quad (\*) $$
valid for $p>1$. Note that Theorem 6.3 in [3] only states the limiting inequality when $n$ tends to infinity on both sides of $(\*)$, but the proof gives $(\*)$ for every finite $n$ as well, since it is a direct application of Cor. 2.2 and Lemma 6.2.
For $p=1$ the inequality (\*) does not hold, one needs an extra log term on the RHS, see (6.6)on page 52 in [3].


Finally, note that a general inequality of the form proposed,
$$\lVert\max\_{n \ge N} \frac{1}{n} \sum\_{i \le n} \phi \circ f^i\rVert\_{L^p} \precsim \lVert\frac{1}{N} \sum\_{i \le N} \phi \circ f^i\rVert\_{L^p} \quad (\*\*) $$
cannot be valid for any $p$. Consider, for example, $f$ to be a rotation of angle $2\pi(\frac{1}{N}-\epsilon)$ on the unit circle, where $\frac{1}{N}-\epsilon $ is irrational. Let $\phi$ be a function that equals 1 on an arc of length $\pi$ on that circle, and equals $-1$ on the complementary arc of the same length. Then for fixed $N>1$, the RHS of $(\*\*)$ tends to zero as $\epsilon \to 0$, while the LHS is at least $1/(N+1)$.
[1] [http://www-stat.wharton.upenn.edu/~steele/Courses/530/Resources/HopfSLLN.pdf](http://www-stat.wharton.upenn.edu/%7Esteele/Courses/530/Resources/HopfSLLN.pdf)


[2] [https://people.math.wisc.edu/~roch/teaching\_files/275b.1.12w/lect15-web.pdf](https://people.math.wisc.edu/%7Eroch/teaching_files/275b.1.12w/lect15-web.pdf)


[3] Krengel, Ulrich. Ergodic theorems. Vol. 6. Walter de Gruyter, 2011.


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