Diophantine approximation and self-conformal measures

It is proved that the Hausdorff measure on the limit set of a finite conformal iterated function system is strongly extremal, meaning that almost all points with respect to this measure are not multiplicatively very well approximable. This proves Conjecture 10.6 from (on fractal measures and Diophantine approximation, preprint, 2003). The strong extremality of all (S,P)-invariant measures is established, where S is a finite conformal iterated function system and P is a probability vector. Both above results are consequences of the much more general Theorem 1.5 concerning Gibbs states of Hölder families of functions.