Dynamical obstruction to the existence of continuous sub-actions for interval maps with regularly varying property

TL;DR

This paper investigates how certain interval maps with regularly varying properties prevent the existence of continuous sub-actions, especially near indifferent fixed points, impacting ergodic optimization techniques.

Contribution

It identifies a class of moduli of continuity incompatible with continuous sub-actions for specific interval maps with regular variation.

Findings

Certain moduli of continuity obstruct sub-actions near fixed points

Results apply to Manneville-Pomeau and Farey maps

Highlights local dynamics' role in ergodic optimization

Abstract

In ergodic optimization theory, the existence of sub-actions is an important tool in the study of the so-called optimizing measures. For transformations with regularly varying property, we highlight a class of moduli of continuity which is not compatible with the existence of continuous sub-actions. Our result relies fundamentally on the local behavior of the dynamics near a fixed point and applies to interval maps that are expanding outside an indifferent fixed point, including Manneville-Pomeau and Farey maps.