We present a panorama of the convergence properties of the 256 Elementary Cellular Automata under fully asynchronous updating, that is, when only one cell is updated at each time step. We regroup here various results which have been presented in different articles and expose a full analysis of the behaviour of finite systems with periodic boundary conditions. Our classification relies on the scaling properties of the average convergence time to a fixed point. We observe that different scaling laws can be found, which fall in one of the following classes: logarithmic, linear, quadratic, exponential and non-converging. The techniques for quantifying this behaviour rely mainly on Markov chain theory and martingales. Most behaviours can be studied analytically but there are still many rules for which obtaining a formal characterisation of their convergence properties is still an open problem.

中文翻译：

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完全异步更新的基本元胞自动机教程

我们提供了在完全异步更新（即每个时间步仅更新一个单元格）下256个​​基本元胞自动机的收敛特性的全景图。我们在这里重新组合了在不同文章中已经介绍过的各种结果，并揭示了具有周期性边界条件的有限系统的行为的完整分析。我们的分类依赖于平均收敛时间到固定点的缩放属性。我们观察到可以找到不同的缩放定律，它们属于以下类别之一：对数，线性，二次，指数和非收敛。量化这种行为的技术主要依赖于马尔可夫链理论和mar。