Typicality of periodic optimization over an expanding circle map

TL;DR

This work addresses ergodic optimization for real-analytic expanding circle maps, proving that for such maps a typical analytic performance function has a unique maximizing measure supported on a periodic orbit. The authors develop a transversality-based framework using calibrated sub-actions, the $(f,T)$-non-wandering set, and a Sturmian-like structure to separate periodic from non-periodic maximizers. They show that non-periodic Sturmian-like maximizing functions form a shy set, while periodic-maximizing behavior is prevalent in natural function spaces, thereby confirming conjectures on the typicality of periodic maximizing measures in this setting. The results advance understanding of TPO for expanding circle maps and provide a robust perturbative method applicable to high-regularity function spaces, with potential implications for related ergodic optimization problems.

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 $r=1,2,\cdots,\infty,ω$.

Theorems & Definitions (53)

• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Theorem 2.4
• Lemma 2.5: CLT01
• proof
• Lemma 2.6