Separation Conditions for Conformal Iterated Function Systems

We extend both the weak separation condition and the finite type condition to include finite iterated function systems (IFSs) of injective C ¹ conformal contractions on compact subsets of ℝ d . For conformal IFSs satisfying the bounded distortion property, we prove that the finite type condition implies the weak separation condition. By assuming the weak separation condition, we prove that the Hausdorff and box dimensions of the attractor are equal and, if the dimension of the attractor is α, then its α-dimensional Hausdorff measure is positive and finite. We obtain a necessary and sufficient condition for the associated self-conformal measure μ to be singular. By using these we give a first example of a singular invariant measure μ that is associated with a non-linear IFS with overlaps.