Suppose a set of prototiles allows $ N $ different substitution rules. In this paper we study tilings of $ \mathbb{R}^d $ constructed from random application of the substitution rules. The space of all possible tilings obtained from all possible combinations of these substitutions is the union of all possible tilings spaces coming from these substitutions and has the structure of a Cantor set. The renormalization cocycle on the cohomology bundle over this space determines the statistical properties of the tilings through its Lyapunov spectrum by controlling the deviation of ergodic averages of the $ \mathbb{R}^d $ action on the tiling spaces.

中文翻译：

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随机替代平铺和偏差现象

假设一组原生动物允许$ N $不同的替代规则。在本文中，我们研究了根据替换规则的随机应用构造的$ \ mathbb {R} ^ d $切片。从这些替换的所有可能组合中获得的所有可能拼贴的空间是来自这些替换的所有可能拼贴的空间的并集，并具有Cantor集的结构。通过控制$ \ mathbb {R} ^ d $作用在平铺空间上的遍历平均数的偏差，在该空间上的同调束上的重整化cocycle通过其Lyapunov谱确定了平铺的统计特性。