Curvature densities of self-similar sets

TL;DR

This paper introduces fractal curvature measures for self-similar sets in Euclidean space, extending classical curvature concepts to fractals using ergodic theory, and relates them to Hausdorff measures.

Contribution

It defines higher order mean curvatures and fractal Gauss-type curvature for self-similar sets, linking them to curvature measures and extending previous global results.

Findings

Fractal curvature measures are proportional to Hausdorff measures on self-similar sets.

The approach uses ergodic theory to analyze local curvature densities.

Global curvature results are extended to a broader class of fractals.

Abstract

For a large class of self-similar sets F in R^d analogues of the higher order mean curvatures of differentiable submanifolds are introduced, in particular, the fractal Gauss-type curvature. They are shown to be the densities of associated fractal curvature measures, which are all multiples of the corresponding Hausdorff measures on F, due to its self-similarity. This local approach based on ergodic theory for an associated dynamical system enables us to extend former global curvature results.