Curvature densities of self-similar sets (Q2851021)

Curvature measures have been introduced for various different set classes using various methods, including convex sets and sets with positive reach. M. Zähle (one of the authors of the present paper) and S. Winter have recently been trying to investigate a second-order fractal ``differential'' geometry and to introduce curvature measures for self-similar sets. In the present paper, the two authors suggest a different approach, a local one. More precisely, for a large class of self-similar sets \(F\) in \(\mathbb{R}^d\) they introduce analogues of the higher-order mean curvatures of differentiable submanifolds, in particular, the fractal Gauss-type curvature. These local fractal curvatures can be interpreted as the densities of associated fractal curvature measures. The main difference in the current approach is that instead of using the renewal theorem from probability theory, Birkhoff's ergodic theorem for an appropriate dynamical system is applied. This technique shortens and simplifies the proofs, as well as allows the authors to extend former total curvature results. It is noteworthy that due to the self-similar structure of \(F\), the curvature measures of all orders are constant multiples of the corresponding Hausdorff measure. At the end, two examples of self-similar sets are provided, the Cantor dust in the plane and the Menger sponge in the space. These examples have been chosen, because the techniques from Zähle and Winter may not be applied but the ones from the current paper are applicable.