We study the two-dimensional continued fraction algorithm introduced by Garrity and the associated triangle map $T$, defined on a triangle $\triangle\subset \mathbb{R}^2$. We introduce a slow version of the triangle map, the map $S$, which is ergodic with respect to the Lebesgue measure and preserves an infinite Lebesgue-absolutely continuous invariant measure. We show that the two maps $T$ and $S$ share many properties with the classical Gauss and Farey maps on the interval, including an analogue of the weak law of large numbers and of Khinchin's weak law for the digits of the triangle sequence, the expansion associated to $T$. Finally, we confirm the role of the map $S$ as a two-dimensional version of the Farey map by introducing a complete tree of rational pairs, constructed using the inverse branches of $S$, in the same way as the Farey tree is generated by the Farey map, and then, equivalently, generated by a generalised mediant operation.

中文翻译：

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带有一段无关不动点和一棵完整的有理对树的慢三角图

我们研究了 Garrity 引入的二维连分数算法和相关的三角形映射 $T$，定义在三角形 $\triangle\subset\mathbb{R}^2$ 上。我们引入了三角形映射的慢速版本，映射 $S$，它相对于 Lebesgue 测度是遍历的，并保留了一个无限的 Lebesgue-绝对连续不变测度。我们表明这两个映射 $T$ 和 $S$ 与区间上的经典高斯和法雷映射共享许多属性，包括大数弱定律和三角形序列数字的 Khinchin 弱定律的类似物，与 $T$ 相关的扩展。最后，我们通过引入一个完整的有理对树，使用 $S$ 的逆分支构造，确认了映射 $S$ 作为 Farey 映射的二维版本的作用，