Fractal curvature measures and Minkowski content for one-dimensional self-conformal sets

TL;DR

This paper investigates the existence and properties of fractal curvature measures and Minkowski content for one-dimensional self-conformal sets, revealing conditions under which these measures exist and their relation to the geometric potential function.

Contribution

It establishes the existence criteria for fractal curvature measures and Minkowski content in conformal iterated function systems, linking them to lattice conditions and conformal measures.

Findings

Fractal curvature measures exist iff the geometric potential is nonlattice.

Minkowski content exists in the nonlattice case and relates to the conformal measure.

In the lattice case, sufficient conditions for Minkowski content existence are provided.

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 and only if the associated geometric potential function is nonlattice. Moreover, in the nonlattice situation we obtain that the Minkowski content exists and prove that the fractal curvature measures are constant multiples of the $\delta$-conformal measure, where $\delta$ 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, contrasting 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 $\mathcal{C}^{1+\alpha}$ diffeomorphic image. Both in the lattice and nonlattice situation average versions of the fractal curvature measures are shown to always exist.