MINIMAX ASPECTS OF OPTIMIZATIONS IN ERGODIC THEORY

The paper’s Theorem 3.2 proves α(φ) = sup_x inf_n (1/n)S_nφ(x) = inf_n sup_x (1/n)S_nφ(x) using a minimax inequality, a standard empirical-measure argument, and an auxiliary Lemma 3.3 that upgrades sup_x lim inf to sup_x inf_n, together with the pointwise ergodic theorem . The candidate solution proves the same identities: the right-hand equality via empirical measures (matching the paper’s approach) and the left-hand equality via a finite Pliss-type selection on maximizing orbit segments (a different, valid route). The arguments are consistent; the paper’s proof is complete modulo a minor tacit step (attainment of the discrete infimum when lim inf is strictly larger), which is standard and readily justified, and the model’s proof correctly uses compactness and continuity to pass to a limit point. Hence both are correct, via different proofs.