Regularity of calibrated sub-actions for circle expanding maps and Sturmian optimization

TL;DR

This paper investigates the regularity of calibrated sub-actions in ergodic optimization for circle expanding maps, showing that near-convex functions have sub-actions close to convex functions, and extends Sturmian measure optimization results.

Contribution

It introduces a simple observation linking near-convex functions to convex sub-actions and generalizes Sturmian measure optimization for circle doubling maps.

Findings

Calibrated sub-actions are closer to convex functions for near-convex functions.

Generalization of Bousch's Sturmian measure optimization result.

Simplified proof avoiding numerical calculations.

Abstract

In this short and elementary note, we study some ergodic optimization problems for circle expanding maps. We first make an observation that if a function is not far from being convex, then its calibrated sub-actions are closer to convex functions in certain effective way. As an application of this simple observation, for circle doubling map, we generalize a result of Bousch saying that translations of the cosine function are uniquely optimized by Sturmian measures. Our argument follows the mainline of Bousch's original proof, while the technical part is simplified by the observation mentioned above, and no numerical calculation is needed.