Duality

Abstract

In this text, we use the dual characterization of the ergodic maximizing value to expose a natural bridge between maximizing probabilities and sub-actions. Since the earliest works in ergodic optimization theory, it became clear however that this constant can be presented in various equivalent ways. In this chapter, some alternative expressions that could be considered to define the ergodic maximizing value are brought to the attention of the reader. Furthermore, a proof of the dual formula is provided.