标题： 几乎处处的遍历平均值的慢收敛 显示英文标题

标题： Slow convergence almost everywhere of ergodic averages

摘要： 对于群$Z^n$在概率空间上的遍历作用，以及给定的任意缓慢趋于零的序列，存在一个可积函数，使得该函数的标准遍历时间平均值几乎处处以不被该序列渐近主导的速率收敛到该函数的空间平均值。 这推广了关于Birkhoff遍历定理中收敛速率不存在普遍估计的Krengel效应。 证明使用了遍历$Z^n$-作用的Rokhlin引理的一个弱化版本。 它确保了具有给定测度的渐近几乎不变集序列的存在性。 这种序列的构造的一个特点是，下一个几乎不变集的选择取决于原函数以及之前不变集的选择。 对于一个正函数的遍历平均值与均值之间的显著偏差，可以在极其长的时间间隔内统一实现。 该偏差可以大于一个正常数，并且在任意长的时间间隔内与之相差很小。 我们不仅能够将指定的偏差任意远地实现，而且这样的偏差序列可以作为任意缓慢消失的序列来实现。 显示英文摘要

摘要： For an ergodic action of the group $Z^n$ on a probability space and a given arbitrarily slowly decreasing to zero sequence, there exists an integrable function such that the standard ergodic time averages for it converge almost everywhere to the spatial average of the function at a rate that is not asymptotically majorized by this sequence. This generalizes the Krengel effect about the absence of universal estimates for the rate of convergence in Birkhoff's ergodic theorem. The proof uses a weakened version of Rokhlin's lemma for ergodic $Z^n$-actions. It ensures the existence of the required sequence of asymptotically almost invariant sets with given measures. A feature of the construction of such a sequence is that the choice of the next almost invariant set depends on the original function and on the choice of the previous invariant sets. A significant deviation of the ergodic averages from the mean of a positive function can be uniformly realized over extremely large time intervals. The deviation can be greater than a positive constant and differ little from it on arbitrary long time intervals. We not only can achieve the specified deviations arbitrarily far, but the sequence of such deviations from the average can be realized as a sequence wanishing arbitrarily slowly.