ABSTRACT Up to words reversal and relabeling, there exists a unique substitution associated with the smallest Pisot number with a minimal number of letters. This is the substitution s: 1↦2, 2↦3, 3↦12. We study the Rauzy fractal of this substitution and show that it is the union of a countable number of Hokkaido tiles and a fractal of dimension strictly less than 2 which is completely explicit. We complete the picture by showing that these Hokkaido tiles are arranged in three different manners to form tiles which are all pairwise disjoint. We also give an efficient algorithm to draw a zoom on a Rauzy fractal. And we show that the symbolic system of the substitution s is measurably isomorphic to a nice domain exchange with pieces. The tools used in this article, using regular languages, are very general and can be easily adapted to study Rauzy fractals of any substitution associated with a Pisot number, and other fractals associated with algebraic numbers without conjugate of modulus one.

中文翻译：

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与最小皮索数相关的最小替代的 Rauzy 分形

摘要 在单词反转和重新标记之前，存在与具有最少字母数的最小 Pisot 数相关的唯一替换。这是替换 s：1↦2、2↦3、3↦12。我们研究了这种替代的 Rauzy 分形，并表明它是可数数量的北海道瓷砖和一个维数严格小于 2 的分形的联合，这是完全明确的。我们通过展示这些北海道瓷砖以三种不同的方式排列以形成所有成对不相交的瓷砖来完成图片。我们还提供了一种有效的算法来在 Rauzy 分形上绘制缩放。并且我们证明了替换 s 的符号系统与碎片的良好域交换是可测量的同构。本文中使用的工具，使用常规语言，