# Discounted Gottschalk-Hedlund theorem

## TL;DR

This paper investigates the convergence of discounted transfer functions in ergodic optimization, establishing that convergence occurs if and only if the observable is balanced, thus linking weak KAM theory and ergodic properties.

## Contribution

It provides a new characterization of convergence for discounted transfer functions in the context of ergodic optimization, specifically for coboundary observables over minimal systems.

## Key findings

- Discounted transfer function converges if and only if the observable is balanced.
- Convergence is characterized within the framework of ergodic optimization and weak KAM theory.
- The result links the convergence property to the balanced nature of the observable.

## Abstract

Ergodic optimization and discrete weak KAM theory are two parallel theories with several results in common. For instance, the Mather set is the locus of orbits which minimize the ergodic averages of a given observable. In the favorable cases, the observable is cohomologous to its ergodic minimizing value on the Mather set, and the discrete weak KAM solution plays the role of the transfer function. One possibility of construction of such a coboundary is by using the non linear Lax-Oleinik operator. The other possibility is by using a discounted cohomological equation. It is known that the discounted discrete weak KAM solution converges to some selected weak KAM solution. We show that, in the ergodic optimization case for a coboundary observable over a minimal system, the discounted transfer function converges if and only if the observable is balanced.