We prove that almost every finite collection of matrices in $GL_d(\mathbb{R})$ and $SL_d(\mathbb{R})$ with positive entries is Diophantine. Next we restrict ourselves to the case $d=2$. A finite set of $SL_2(\mathbb{R})$ matrices induces a (generalized) iterated function system on the projective line $\mathbb{RP}^1$. Assuming uniform hyperbolicity and the Diophantine property, we show that the dimension of the attractor equals the minimum of 1 and the critical exponent.

中文翻译：

---

ℝℙ1中射影迭代函数系统的矩阵和吸引子的丢番图性质

我们证明了 $GL_d(\mathbb{R})$ 和 $SL_d(\mathbb{R})$ 中几乎所有具有正项的矩阵的有限集合都是丢番图。接下来我们将自己限制在 $d=2$ 的情况下。一组有限的 $SL_2(\mathbb{R})$ 矩阵在投影线 $\mathbb{RP}^1$ 上归纳出一个（广义）迭代函数系统。假设均匀双曲线和丢番图性质，我们表明吸引子的维数等于 1 和临界指数的最小值。