Typicality of periodic optimization over an expanding circle map

TL;DR

This paper investigates the typical behavior of ergodic optimization on expanding circle maps, showing that most performance functions have a unique periodic maximizing measure, both topologically and measure-theoretically.

Contribution

It proves that for a broad class of smooth functions, the maximizing measure is generically unique and supported on a periodic orbit in the context of expanding circle maps.

Findings

Most performance functions have a unique periodic maximizing measure.

The unique maximizing measure is supported on a periodic orbit.

Results hold for functions with various degrees of smoothness, including real analytic.

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 $r=1,2,\cdots,\infty,\omega$.