Abstract
Given any ergodic invertible measure preserving transformation τ of [0,1] and any null-sequence (α N ) of positive reals, there exists a continuousf such that
i.e. there is no “speed of convergence” in the ergodic theorem for any τ. The analogous result holds also for norm-convergence.
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References
Friedman, N. A.: Introduction to Ergodic Theory. New York-Cincinnati-Toronto-London-Melbourne: Van Nostrand. 1970.
Halmos, P. A.: Lectures on Ergodic Theory. Tokyo: The Math. Soc. Japan. 1956.
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Krengel, U. On the speed of convergence in the ergodic theorem. Monatsh Math 86, 3–6 (1978). https://doi.org/10.1007/BF01300052
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DOI: https://doi.org/10.1007/BF01300052