Geodesic
The maximum pointwise rate of convergence in Birkhoff's ergodic theorem
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXXI, Tome 498 (2020), pp. 18-25 
A. G. Kachurovskii; I. V. Podvigin; A. A. Svishchev. The maximum pointwise rate of convergence in Birkhoff's ergodic theorem. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXXI, Tome 498 (2020), pp. 18-25. http://geodesic.mathdoc.fr/item/ZNSL_2020_498_a1/
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     title = {The maximum pointwise rate of convergence in {Birkhoff's} ergodic theorem},
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

A criterion for the maximum possible pointwise convergence rate in Birkhoff's ergodic theorem for ergodic semiflows in a Lebesgue space is obtained. It is proved that higher rates of convergence in this theorem are impossible.

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