Ergodic Optimization

Let f be a real-valued function defined on the phase space of a dynamical system. Ergodic optimization is the study of those orbits, or invariant probability measures, whose ergodic f -average is as large as possible. In these notes we establish some basic aspects of the theory: equivalent definitions of the maximum ergodic average, existence and generic uniqueness of maximizing measures, and the fact that every ergodic measure is the unique maximizing measure for some continuous function. Generic properties of the support of maximizing measures are described in the case where the dynamics is hyperbolic. A number of problems are formulated.