On limiting measures for a class of one-dimensional linear cellular automata
On limiting measures for a class of one-dimensional linear cellular automata
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关于一类一维线性元胞自动机的极限测度
DOI：
10.1109/candar.2016.0049
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发表时间：
2016
期刊：
影响因子：
--
通讯作者：
Masato Takei
中科院分区：
文献类型：
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作者：
Ryouta Kouduma;Masato Takei;Masato Takei
线性细胞自动机一般有很多不变测度，但最自然的不变测度是一致伯努利积测度。关于它们的刚性有几个研究：唯一不变测度，具有适当的非退化条件（如正熵或移位映射的混合性质），是一致测度。这与渐近随机化性质的研究有关：从一大类初始测度开始的迭代收敛到一致测度（在塞萨罗意义下）。本文考虑邻域大小为2的一维线性元胞自动机，从一类平移不变的概率测度出发，研究极限分布。我们的特点，当迭代由除了模一个素数元胞自动机开始从一个强混合概率测度与全支持可以收敛。这也给出了这些概率测度类内的所有不变测度。在两状态情形下，我们也得到了强混合概率测度的凸组合在模2加法元胞自动机下不变的一个充要条件。这些结果改进了Marcovici和Miyamoto的结果。
Linear cellular automata have many invariant measures in general, but the most natural one is the uniform Bernoulli product measure. There are several studies on their rigidity: The unique invariant measure with a suitable non-degeneracy condition (such as positive entropy or mixing property for the shift map) is the uniform measure. This is related to study of the asymptotic randomization property: Iterates starting from a large class of initial measures converge to the uniform measure (in Cesaro sense). In this paper we consider one-dimensional linear cellular automata with neighborhood of size two, and study limiting distributions starting from a class of shift-invariant probability measures. We characterize when iterates by addition modulo a prime number cellular automata starting from a strong mixing probability measure with full support can converge. This also gives all invariant measures inside the class of those probability measures. In the two-state case, we also obtain a necessary and sufficient condition that a convex combination of strong mixing probability measures is invariant under addition modulo 2 cellular automata. Those results improve previous ones obtained by Marcovici and Miyamoto.