Adversarial ergodic optimization

In this article, we introduce an ergodic optimization problem inspired by information theory, which can be presented informally as follows: given a factor map $\pi :(X,T)\to (Y,S)$ of topological dynamical systems, and a continuous function $f\in C(X)$, what can be said about the extrema $$\underset{y\in Y}{\sup }\underset{x\in {\pi }^{-1}\{y\}}{\inf }\underset{k\to \mathrm{\infty }}{\lim }\frac{1}{k}\sum _{j=0}^{k-1}f\left(T^{j}x\right)?$$