Minimax aspects of optimizations in ergodic theory

TL;DR

This paper explores minimax optimization problems in ergodic theory, providing characterizations of ergodic averages and linking them to duality principles, thereby advancing theoretical understanding.

Contribution

It introduces minimax characterizations of ergodic averages and connects these results to the Fenchel-Rockafellar duality framework, extending the variational principles in ergodic theory.

Findings

Provided minimax characterizations of maximum ergodic averages.

Linked minimax results to Fenchel-Rockafellar duality.

Extended the variational principle for generalized pressure functions.

Abstract

We study optimization problems in ergodic theory from the view point of minimax problems. We give minimax characterizations of maximum ergodic averages involving time averages. Our approach works for the abstract variational principle of generalized pressure functions which is proved by Bi\'{s} et al. (2022). We also describe the relationship between our minimax results and the Fenchel-Rockafellar duality.