Fractal Measures for Parabolic IFS

AbstractLet h be the Hausdorff dimension of the limit set of a conformal parabolic iterated function system in dimension d⩾2. In case the system of maps is finite, we provide necessary and sufficient conditions for the h-dimensional Hausdorff measure to be positive and finite and also, assuming the strong open set condition holds, characterize when the h-dimensional packing measure of the limit set is positive and finite. We also prove that the upper ball (box)-counting dimension and the Hausdorff dimension of this limit set coincide. As a byproduct we include a compact analysis of the behaviour of parabolic conformal diffeomorphisms in dimension 2 and separately in any dimension greater than or equal to 3