Absolute continuity and transversality for fractional linear IFS's

In my talk, the main question is: When will the self-similar measure of an IFS be absolute continuous? This question is not answered exactly. It is the general case of Erdos's question, which is for linear functions with one parameter. There are theorems, which answer Erdos's question. But in more difficult cases we do not have the exact answer. However there are some theorems which help us to get a set of parameters, where the invariant measure is absolute continuous almost everywhere. Those theorems use an important condition, namely transversality condition which was introduced by Karoly Simon and Mark Pollicott. I will use this condition in a special case of IFSs where the functions are fractional linear. I will show, how can we prove in this case that the IFS satisfies the transversality condition, and how can we use the theorem of Simon, Solomyak and Urbanski to prove absolute continuity.