On the speed of convergence in the ergodic theorem

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

$$\lim \sup \alpha _{\rm N}^{ - 1} \left| {N^{ - 1} \sum\limits_{k = 0}^{N - 1} {f \circ \tau ^k - \smallint f} } \right| = \infty a. e.,$$

i.e. there is no “speed of convergence” in the ergodic theorem for any τ. The analogous result holds also for norm-convergence.