Diophantine Property of Matrices and Attractors of Projective Iterated Function Systems in ℝℙ¹

We prove that almost every finite collection of matrices in GLd (ℝ) and SLd({ℝ) with positive entries is Diophantine. Next we restrict ourselves to the case d=2. A finite set of SL2(ℝ) matrices induces a (generalized) iterated function system on the projective line ℝℙ1. Assuming uniform hyperbolicity and the Diophantine property, we show that the dimension of the attractor equals the minimum of 1 and the critical exponent.