Ergodic optimization in dynamical systems

Abstract

Ergodic optimization is the study of problems relating to maximizing orbits and invariant measures, and maximum ergodic averages. An orbit of a dynamical system is called$f$-maximizing if the time average of the real-valued function$f$along the orbit is larger than along all other orbits, and an invariant probability measure is called$f$-maximizing if it gives$f$a larger space average than any other invariant probability measure. In this paper, we consider the main strands of ergodic optimization, beginning with an influential model problem, and the interpretation of ergodic optimization as the zero temperature limit of thermodynamic formalism. We describe typical properties of maximizing measures for various spaces of functions, the key tool of adding a coboundary so as to reveal properties of these measures, as well as certain classes of functions where the maximizing measure is known to be Sturmian.