Dimension of attractors for Iterated Function Systems of linear fractional transformations and the Diophantine property of matrix semigroups

We consider Iterated Function Systems of linear fractional transformations, and show that the
Hausdorff dimension of the attractor is given by the Bowen's pressure formula, if the Iterated
Function Systems satisfy the exponential separation condition. We also show that almost every
finite collections of $GL_n( \mathbb{R} )$ matrices are Diophantine if the matrices have positive
entries. This is a joint work with Boris Solomyak.