We show that for any countable amenable group action, along certain Følner sequences (those that have for any $c>1$ a two-sided $c$-tempered tail), one has a universal estimate for the number of fluctuations in the ergodic averages of $L^{\infty}$ functions. This estimate gives exponential decay in the number of fluctuations. Any two-sided Følner sequence can be thinned out to satisfy the above property. In particular, any countable amenable group admits such a sequence. This extends results of S. Kalikow and B. Weiss [1] for $\mathbb{Z}^{d}$ actions and of N. Moriakov [3] for actions of groups with polynomial growth.

中文翻译：

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适合群体行动的遍历平均值的波动

我们表明，对于任何可数的服从群体行动，沿着某些 Følner 序列（那些对任何 $c>1$ 有一个双面 $c$-tempered 尾巴的序列），人们对遍历中的波动数量有一个普遍的估计$L^{\infty}$ 函数的平均值。这个估计给出了波动数量的指数衰减。任何两侧的 Følner 序列都可以被稀疏化以满足上述性质。特别是，任何可数的服从组都承认这样的序列。这扩展了 S. Kalikow 和 B. Weiss [1] 的 $\mathbb{Z}^{d}$ 动作和 N. Moriakov [3] 的多项式增长组动作的结果。