The maximum ergodic average α(φ) equals the supremum over points of the infimum over time of Birkhoff averages, and also equals the infimum over time of the supremum over points.
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Minimax aspects of optimizations in ergodic theory
The maximum ergodic average α(φ) equals the supremum over points of the infimum over time of Birkhoff averages, and also equals the infimum over time of the supremum over points.