Diophantine property of matrices and attractors of projective iterated function systems in $\mathbb{RP}^1$

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.
Comment: 26 pages; small changes in the proof of Theorem 1.7 compared with the previous version. Accepted for publication in IMRN