Typicality of periodic optimization over an expanding circle map

Abstract

We study the ergodic optimization problem over a real analytic expanding circle map. We show that in both the topological and the measure-theoretical senses, a typical $C^r$ performance function has a unique maximizing measure and the unique maximizing measure is supported on a periodic orbit, for $r=1,2,\dots ,\infty ,\ddot{I}‰$.

30 pages, minor corrections