Hi. I've been stuck on the following question for some time.


Consider a sequence of functions $\left( f\_n \right)$ from an ergodic space $\left( \mathsf{X}, \mathsf{S}, \mu \right)$ to $\left[ 0,1 \right]$ such that $f\_{a+b} \leq f\_a \mathsf{S}^a \left( f\_b \right)$ for all integers $a, b \geq 0$, where, classically, $\mathsf{S} \left( g \right)$ is the map $x \longmapsto g \left( \mathsf{S} \left( x \right) \right)$.


Obviously $f\_n \leq f\_{n+1}$ so that the sequence $\left( f\_n \right)$ decreases at each point to a function $f$. Under the hypothesis $\int\_{\mathsf{X}} f\_1 d\mu < 1$ I was able to prove that $f = 0$ almost everywhere. Here's how I dealt with the problem:


For fixed $n$ and integers $k$ and $\alpha$ such that $k \alpha \leq n$, the inequality
\begin{equation}
f\_n \leq f\_1 \mathsf{S}^k \left( f\_1 \right) \ldots \mathsf{S}^{k\alpha} \left( f\_1 \right)
\end{equation} 
holds, so that, taking averages, one gets
\begin{equation}
f\_n \leq \frac{1}{\lfloor n/ \alpha \rfloor} \sum\_{k=0}^{\lfloor n/ \alpha \rfloor} f\_1 \mathsf{S}^k \left( f\_1 \right) \ldots \mathsf{S}^{k\alpha} \left( f\_1 \right)
\end{equation}
Taking integrals, and using the dominated convergence theorem and the Furstenberg-Katznelson theorem on multiple ergodic averages, one gets
\begin{equation}
\int\_{\mathsf{X}} f \leq lim\_{n \longrightarrow \infty} \frac{1}{\lfloor n/ \alpha \rfloor} \sum\_{k=0}^{\lfloor n/ \alpha \rfloor} \int\_{\mathsf{X}} f\_1 \mathsf{S}^k \left( f\_1 \right) \ldots \mathsf{S}^{k\alpha} \left( f\_1 \right) = \left( \int\_{\mathsf{X}} f\_1 \right)^\alpha
\end{equation}
For all integers $\alpha$, which allows me to conclude.


Now the question is: does the series $\sum f\_n$ converge? That seems plausible considering the seemingly exponential decreasing of $f\_n$ to $f$, but trying to use the same techniques leads to multiple ergodic averages for a varying number of terms - making the integer $\alpha$ dependent on $n$ that is; is there any way to deal with those? All suggestions are welcome.



Multiple ergodic averages with varying number of terms
Unless I miss something, you are overcomplicating the story. In the first place, as it has already been pointed out, you don't really need submultiplicativity as 
$$
f\_n(x) \le f\_1(x) f\_1(Sx) \dots f\_1(S^{n-1} x) = F\_n(x) \;.
$$
Applying the usual ergodic theorem to the right-hand side one gets that a.e.
$$
\log F\_n(x)/n \to \int\log f\_1(x)\;d\mu(x) < 0 \;,
$$
whence the claim.


