Fractal curvature measures and Minkowski content for self-conformal subsets of the real line

Abstract

We show that the fractal curvature measures of invariant sets of one-dimensional conformal iterated function systems satisfying the open set condition exist if the associated geometric potential function is nonlattice. Moreover, we prove that if the maps of the conformal iterated function system are all analytic, then the fractal curvature measures do not exist in the lattice case. Further, in the nonlattice situation we obtain that the Minkowski content exists and prove that the fractal curvature measures are constant multiples of the δ-conformal measure, where δ denotes the Minkowski dimension of the invariant set. For the first fractal curvature measure, this constant factor coincides with the Minkowski content of the invariant set. In the lattice situation we give sufficient conditions for the Minkowski content of the invariant set to exist, disproving a conjecture of Lapidus and standing in contrast with the fact that the Minkowski content of a self-similar lattice fractal never exists. However, every self-similar set satisfying the open set condition exhibits a Minkowski measurable C1+α diffeomorphic image. Both in the lattice and in the nonlattice situation, average versions of the fractal curvature measures are shown to always exist.