In this joint work in progress with Crovisier, Poletti, Tahzibi, we consider Discretized Anosov Flows (eg, perturbations of time-one maps of Anosov flows) and study their measures of maximal entropy (MMEs) under some irreducibility condition. We prove the following dichotomy: either (1) there are exactly two ergodic  MMEs, one with positive and one with negative center exponent, or (2) there is a unique MME with zero center exponent. Moreover, the case (1) is generic and implies exponential mixing for each ergodic MME.