Ergodic Optimization

The field is a relatively recently established subfield of ergodic theory, and has significant input from the two well-established areas of symbolic dynamics and Lagrangian dynamics. The large-scale picture of the field is that one is interested in optimizing potential functions over the (typically highly complex) class of invariant measures for a dynamical system. Tools that have been employed in this area come from convex analysis, statistical physics, probability theory and dynamic programming. The field also has both general aspects (in which the optimization is considered in the large on whole Banach spaces) and local aspects (in which the optimization is studied on individual functions). In the latter category, there has been input from physicists with numerical simulations suggesting that the optimizing measures are typically supported on periodic orbits. This should be contrasted with the situation typically found in the ‘thermodynamic formalism’ of ergodic theory, in which the measures picked out by variational principles tend to have wide support. Ergodic optimization may be viewed as the low temperature limit of thermodynamic formalism.