Curvature-direction measures of self-similar sets

TL;DR

This paper introduces fractal curvature-direction measures for self-similar sets, providing a way to analyze their geometric structure by combining normal vector distribution and curvature localization, using ergodic theory.

Contribution

It develops a new method to compute curvature-direction measures for self-similar fractals, extending classical geometric concepts to fractal sets with minimal assumptions.

Findings

Measures describe normal vector distribution and curvature localization.

Measures decouple into product of Hausdorff measure and a fiber measure.

Integral formula for the fiber measure enables practical computation.

Abstract

We obtain fractal Lipschitz-Killing curvature-direction measures for a large class of self-similar sets F in R^d. Such measures jointly describe the distribution of normal vectors and localize curvature by analogues of the higher order mean curvatures of differentiable submanifolds. They decouple as independent products of the unit Hausdorff measure on F and a self-similar fibre measure on the sphere, which can be computed by an integral formula. The corresponding local density approach uses an ergodic dynamical system formed by extending the code space shift by a subgroup of the orthogonal group. We then give a remarkably simple proof for the resulting measure version under minimal assumptions.