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

Part of: Low-dimensional dynamical systems Ergodic theory

Published online by Cambridge University Press:  27 April 2022

RUI GAO
RUI GAO*
Affiliation:
College of Mathematics, Sichuan University, Chengdu 610064, China
*
e-mail: gaoruimath@scu.edu.cn
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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 a certain effective way. As an application of this simple observation, for a 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 some technical part is simplified by the observation mentioned above, and no numerical calculation is needed.

Keywords

ergodic optimizationcalibrated sub-actioncircle expanding mapSturmian measure

MSC classification

Primary: 37A05: Measure-preserving transformations
Secondary: 37E10: Maps of the circle

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 6 , June 2023 , pp. 1909 - 1921
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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