Mathematical Research Letters

Volume 28 (2021)

Number 1

Gottschalk–Hedlund theorem revisited

Pages: 285 – 300



Xifeng Su (Laboratory of Mathematics and Complex Systems, School of Mathematical Sciences, Beijing Normal University, Beijing, China)

Philippe Thieullen (Institut de Mathématiques de Bordeaux, Université de Bordeaux, Talence, France)


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.

X. Su is supported by the National Natural Science Foundation of China (Grant No. 11971060, 11871242).

Received 22 January 2019

Accepted 31 July 2019

Published 24 May 2022