Mixed ergodic optimization

In this article, we introduce an ergodic optimization problem inspired by information theory, which can be presented informally as follows: given topological dynamical systems (X, T), (Y, ), and a continuous function f C (X Y), what can be said about the extrema ₘ ₘ ₗ ₗ ₊ 1k ₉ = ₀^k - 1 f (Tʲ x, Sʲ y)? Our goal in this article is to formalize this problem, present some preliminary results on the topic, and note some remaining open questions in the general theory.