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

Abstract. We investigate convergence of sequences of n × n matrix exponential functions t → etA k for t > 0, where Ak → A, Ak is nonsingular and A is nilpotent. Specifically, we address pointwise convergence, almost uniform convergence, and, viewing the exponential as a Schwartz distribution, weak∗ convergence. We show that simple results can be obtained in terms of the eigenvalues of A−1 k alo...