Introduction to expanding ergodic optimization

These lecture notes grew out of a graduate course on ergodic optimization given by the author at the University of Campinas. Obviously some back-ground in ergodic theory is required to follow the text. Moreover, these notes are by no means meant to be exhaustive. As a matter of fact, we focus mostly on the interpretation of ergodic optimal problems as questions of variational dynamics (see, for instance, [30, 37, 38, 55]), in a compara-ble way to the Aubry-Mather theory for Lagrangian systems. The reader shall be conscious that other points of view are also useful in ergodic op-timization, like the one based on properties of Sturmian measures and its generalizations (see, for example, [14, 21, 48]). Ergodic optimization is a theoretical branch primarily concerned with the study of the so-called optimizing probability measures. The goal of this introductory monograph is hence twofold. One objective is to present and