Abstract

Birkhoff’s theorem asserts that, for an ergodic automorphism, time averages converge to the space average. Krengel showed that, for a given sequence \(\psi(n)\to+0\) and any ergodic automorphism, there exists an indicator function such that the corresponding time means converge a.e. slower than \(\psi\). We give a new proof of the absence of estimates for rates of convergence, answering a question of Podvigin.

中文翻译：

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遍历平均值的缓慢收敛

摘要

伯克霍夫定理断言，对于遍历自同构，时间平均值收敛于空间平均值。Krengel 表明，对于给定序列\(\psi(n)\to+0\)和任何遍历自同构，存在一个指示函数，使得相应的时间均值收敛 ae 慢于\(\psi\)。我们给出了缺乏收敛率估计的新证明，回答了 Podvigin 的问题。