The average density of self-conformal measures

The paper calculates the average density of the normalized Hausdorff measur e on the fractal set generated by a conformal iterated function system. It equals almost everywhere a positive constant given by a truncated generaliz ed s-energy integral, where s is the corresponding Hausdorff dimension. As a main tool a conditional Gibbs measure is determined. The appendix proves an appropriate extension of Birkhoffs ergodic theorem which is also of inde pendent interest.