A map $f: X \to X$ preseves an ergodc probability $\mu$, i.e., $\mu \circ f^{-1}=\mu$ and if 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

$$||\max_{n \ge N} \frac{1}{n} \sum_{i \le n} \phi \circ f^i||_{L^1} \precsim ||\frac{1}{N} \sum_{i \le N} \phi \circ f^i||_{L^1} \text{ ?}$$