Scaling exponents of curvature measures

  • Dušan Pokorný

    Charles University Prague, Czech Republic
  • Steffen Winter

    Karlsruhe Institute of Technology, Germany

Abstract

Fractal curvatures of a set FRdF\subset\mathbb R^d are roughly defined as suitably rescaled limits of the total curvatures of its parallel sets FϵF_\epsilon as ϵ\epsilon tends to 00 and have been studied in the last years in particular for self-similar and self-conformal sets. This previous work was focussed on establishing the existence of (averaged) fractal curvatures and related fractal curvature measures in the generic case when the curvature measures rescale like the volume of the parallel sets, that is, when the associated scaling exponents coincide with the Minkowski dimension of FF. In the present paper we study the nongeneric situation when the scaling exponents do not coincide with the dimension. We demonstrate that the possibilities for nongeneric behaviour are rather limited and introduce the notion of local flatness, which allows a geometric characterization of nongenericy in R\mathbb R and R2\mathbb R^2. We expect local flatness to be characteristic also in higher dimensions. The results enlighten the geometric meaning of the scaling exponents.

Cite this article

Dušan Pokorný, Steffen Winter, Scaling exponents of curvature measures. J. Fractal Geom. 1 (2014), no. 2, pp. 177–219

DOI 10.4171/JFG/5